| L(s) = 1 | + 12·4-s + 20·11-s + 80·16-s + 226·19-s + 440·29-s − 378·31-s − 260·41-s + 240·44-s + 686·49-s + 1.12e3·59-s + 458·61-s + 192·64-s + 1.78e3·71-s + 2.71e3·76-s + 54·79-s + 1.50e3·89-s − 3.00e3·101-s + 1.21e3·109-s + 5.28e3·116-s − 2.36e3·121-s − 4.53e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.548·11-s + 5/4·16-s + 2.72·19-s + 2.81·29-s − 2.19·31-s − 0.990·41-s + 0.822·44-s + 2·49-s + 2.47·59-s + 0.961·61-s + 3/8·64-s + 2.97·71-s + 4.09·76-s + 0.0769·79-s + 1.78·89-s − 2.95·101-s + 1.06·109-s + 4.22·116-s − 1.77·121-s − 3.28·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.634543851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.634543851\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{2} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2006 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9777 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 113 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17773 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 220 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 189 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 72406 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 130 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158914 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 182046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 100407 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 560 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 229 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39026 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 890 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 27 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 959533 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 750 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 365054 T^{2} + p^{6} T^{4} \) |
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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
−10.31100324848719883772596310311, −9.974285234857571612026666419788, −9.496915678336895127381620007259, −9.144316958133109146501748389677, −8.436911828076371933487949925568, −8.155621266457682224941099054475, −7.54038180731919119104274126039, −7.14244796215259338992014450778, −6.72116379195482115827311202346, −6.66774978043963571993070967693, −5.69376859870243811631358552347, −5.48339275306930437204098102593, −5.05182237100380951611648609313, −4.20424765593468196328515851235, −3.45305088702576324831014540558, −3.29326343535137960601161293335, −2.45572955818027593524656405441, −2.08738125990028598854785370066, −1.11159248792518270497332993930, −0.849938154992172325223707643723,
0.849938154992172325223707643723, 1.11159248792518270497332993930, 2.08738125990028598854785370066, 2.45572955818027593524656405441, 3.29326343535137960601161293335, 3.45305088702576324831014540558, 4.20424765593468196328515851235, 5.05182237100380951611648609313, 5.48339275306930437204098102593, 5.69376859870243811631358552347, 6.66774978043963571993070967693, 6.72116379195482115827311202346, 7.14244796215259338992014450778, 7.54038180731919119104274126039, 8.155621266457682224941099054475, 8.436911828076371933487949925568, 9.144316958133109146501748389677, 9.496915678336895127381620007259, 9.974285234857571612026666419788, 10.31100324848719883772596310311
