| L(s) = 1 | − 2·2-s − 20·4-s − 40·8-s − 976·11-s − 3.28e3·16-s − 4.16e3·17-s − 1.45e3·19-s + 1.95e3·22-s + 1.93e4·25-s + 1.82e4·32-s + 8.33e3·34-s + 2.91e3·38-s + 1.17e5·41-s + 1.97e5·43-s + 1.95e4·44-s + 2.36e5·49-s − 3.86e4·50-s − 5.42e5·59-s + 1.29e5·64-s − 7.90e5·67-s + 8.33e4·68-s + 4.43e5·73-s + 2.91e4·76-s − 2.35e5·82-s + 3.46e6·83-s − 3.94e5·86-s + 3.90e4·88-s + ⋯ |
| L(s) = 1 | − 1/4·2-s − 0.312·4-s − 0.0781·8-s − 0.733·11-s − 0.800·16-s − 0.848·17-s − 0.212·19-s + 0.183·22-s + 1.23·25-s + 0.557·32-s + 0.212·34-s + 0.0530·38-s + 1.71·41-s + 2.48·43-s + 0.229·44-s + 2.00·49-s − 0.308·50-s − 2.63·59-s + 0.492·64-s − 2.62·67-s + 0.265·68-s + 1.14·73-s + 0.0663·76-s − 0.427·82-s + 6.05·83-s − 0.620·86-s + 0.0572·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.507486441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.507486441\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $D_{4}$ | \( 1 + p T + 3 p^{3} T^{2} + p^{7} T^{3} + p^{12} T^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 772 p^{2} T^{2} + 2049246 p^{3} T^{4} - 772 p^{14} T^{6} + p^{24} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 236356 T^{2} + 28021959366 T^{4} - 236356 p^{12} T^{6} + p^{24} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 488 T + 2238078 T^{2} + 488 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 4994596 T^{2} + 30260873415846 T^{4} - 4994596 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2084 T + 32560902 T^{2} + 2084 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 728 T + 92449758 T^{2} + 728 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 212117956 T^{2} + 1001159632705962 p T^{4} - 212117956 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 1409719204 T^{2} + 1160515330165289766 T^{4} - 1409719204 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 2758360324 T^{2} + 3362667870952277766 T^{4} - 2758360324 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 8121202276 T^{2} + 29600495645847907686 T^{4} - 8121202276 p^{12} T^{6} + p^{24} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 58972 T + 240432438 p T^{2} - 58972 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 2296 p T + 13190101374 T^{2} - 2296 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 13578767236 T^{2} + \)\(16\!\cdots\!86\)\( T^{4} - 13578767236 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 42194545636 T^{2} + \)\(11\!\cdots\!86\)\( T^{4} - 42194545636 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 271016 T + 102678575166 T^{2} + 271016 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 2637195124 p T^{2} + \)\(11\!\cdots\!46\)\( T^{4} - 2637195124 p^{13} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 395096 T + 204987839262 T^{2} + 395096 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 164364621124 T^{2} + \)\(21\!\cdots\!06\)\( T^{4} - 164364621124 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 221956 T + 284381984742 T^{2} - 221956 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 828257339524 T^{2} + \)\(28\!\cdots\!06\)\( T^{4} - 828257339524 p^{12} T^{6} + p^{24} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1732504 T + 1400265667422 T^{2} - 1732504 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 380612 T + 970422538278 T^{2} + 380612 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 463388 T + 953987784774 T^{2} + 463388 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.362654962318828889631596875615, −9.284244917000013680981393622307, −9.193703838892524963755499583428, −8.685417759548323785431873357980, −8.500264667298729194682953033656, −7.79802744147103975103682721718, −7.73320170406000030097223406242, −7.69956827531649231233559295947, −6.88852544556208397546520324226, −6.81476543583410742597067926383, −6.45269365035294281883050051230, −5.85868771933313690022961960491, −5.84913819229800943728098728800, −5.20072343632846941162200646060, −4.93325944306615519389551693564, −4.46376348222336564306849012733, −4.09466344967947526616364491680, −3.99391701855247016380248175575, −3.06607096309906918777388836265, −2.72862168625655847689505665458, −2.49627005421609508203295178069, −1.94056171825976796535114757984, −1.28080777415060509583265463376, −0.56600724290255138206839217556, −0.47686578289209351033506776739,
0.47686578289209351033506776739, 0.56600724290255138206839217556, 1.28080777415060509583265463376, 1.94056171825976796535114757984, 2.49627005421609508203295178069, 2.72862168625655847689505665458, 3.06607096309906918777388836265, 3.99391701855247016380248175575, 4.09466344967947526616364491680, 4.46376348222336564306849012733, 4.93325944306615519389551693564, 5.20072343632846941162200646060, 5.84913819229800943728098728800, 5.85868771933313690022961960491, 6.45269365035294281883050051230, 6.81476543583410742597067926383, 6.88852544556208397546520324226, 7.69956827531649231233559295947, 7.73320170406000030097223406242, 7.79802744147103975103682721718, 8.500264667298729194682953033656, 8.685417759548323785431873357980, 9.193703838892524963755499583428, 9.284244917000013680981393622307, 9.362654962318828889631596875615
