| L(s) = 1 | − 14·5-s − 40·11-s − 168·19-s + 71·25-s − 12·29-s − 448·31-s − 532·41-s + 610·49-s + 560·55-s + 56·59-s + 364·61-s − 816·71-s + 96·79-s + 3.05e3·89-s + 2.35e3·95-s − 2.49e3·101-s + 1.80e3·109-s − 1.46e3·121-s + 756·125-s + 127-s + 131-s + 137-s + 139-s + 168·145-s + 149-s + 151-s + 6.27e3·155-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 1.09·11-s − 2.02·19-s + 0.567·25-s − 0.0768·29-s − 2.59·31-s − 2.02·41-s + 1.77·49-s + 1.37·55-s + 0.123·59-s + 0.764·61-s − 1.36·71-s + 0.136·79-s + 3.63·89-s + 2.54·95-s − 2.45·101-s + 1.58·109-s − 1.09·121-s + 0.540·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.0962·145-s + 0.000549·149-s + 0.000538·151-s + 3.25·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3269654977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3269654977\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 14 T + p^{3} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1658 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4962 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 212 T + p^{3} T^{2} )( 1 + 212 T + p^{3} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 86410 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 65914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67122 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 163690 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 182 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 419050 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 390542 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1103370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1526 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1514050 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.57037507037630642639808673273, −11.77669818905802140565405942860, −11.65295573273634771471050131895, −10.71476298187480705420149015447, −10.70706038648466942399381255266, −10.21299893672745905977942253761, −9.325208600454163039691562291832, −8.638543572995007687784064276782, −8.565891132663921558044551008953, −7.61688440512736151183570116196, −7.55684939264998672527267251737, −6.79046717360938480780108511116, −6.16690426301636801334512931406, −5.33131951298514403440543197000, −4.90353106567594370655770268257, −3.89089684308246381525287739215, −3.77288117294267967250499455523, −2.64469510363550218822410407221, −1.84732619274519595504480590811, −0.24733294513370246539893346730,
0.24733294513370246539893346730, 1.84732619274519595504480590811, 2.64469510363550218822410407221, 3.77288117294267967250499455523, 3.89089684308246381525287739215, 4.90353106567594370655770268257, 5.33131951298514403440543197000, 6.16690426301636801334512931406, 6.79046717360938480780108511116, 7.55684939264998672527267251737, 7.61688440512736151183570116196, 8.565891132663921558044551008953, 8.638543572995007687784064276782, 9.325208600454163039691562291832, 10.21299893672745905977942253761, 10.70706038648466942399381255266, 10.71476298187480705420149015447, 11.65295573273634771471050131895, 11.77669818905802140565405942860, 12.57037507037630642639808673273
