| L(s) = 1 | + 33.0·2-s + 582.·4-s − 625·5-s − 2.40e3·7-s + 2.34e3·8-s − 2.06e4·10-s + 2.11e4·11-s + 2.00e4·13-s − 7.94e4·14-s − 2.20e5·16-s + 6.33e5·17-s − 1.62e5·19-s − 3.64e5·20-s + 7.01e5·22-s − 5.02e5·23-s + 3.90e5·25-s + 6.62e5·26-s − 1.39e6·28-s + 5.70e6·29-s − 8.06e6·31-s − 8.50e6·32-s + 2.09e7·34-s + 1.50e6·35-s − 1.35e6·37-s − 5.36e6·38-s − 1.46e6·40-s − 1.81e7·41-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 1.13·4-s − 0.447·5-s − 0.377·7-s + 0.202·8-s − 0.653·10-s + 0.436·11-s + 0.194·13-s − 0.552·14-s − 0.842·16-s + 1.83·17-s − 0.285·19-s − 0.509·20-s + 0.638·22-s − 0.374·23-s + 0.200·25-s + 0.284·26-s − 0.430·28-s + 1.49·29-s − 1.56·31-s − 1.43·32-s + 2.69·34-s + 0.169·35-s − 0.118·37-s − 0.417·38-s − 0.0904·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 - 33.0T + 512T^{2} \) |
| 11 | \( 1 - 2.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.00e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.33e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.62e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.02e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.70e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.06e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.35e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.81e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.87e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.62e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.15e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.80e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.79e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.70e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.95e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.762639159623929431209071468560, −8.608062959824889940772216486756, −7.44795713616238535408016937895, −6.43269473648641142762145087041, −5.60325304855538044867513812108, −4.60760790607431063486251286581, −3.60499554580727396893945834866, −3.01065370314627916851047811621, −1.48374305578700256848232013494, 0,
1.48374305578700256848232013494, 3.01065370314627916851047811621, 3.60499554580727396893945834866, 4.60760790607431063486251286581, 5.60325304855538044867513812108, 6.43269473648641142762145087041, 7.44795713616238535408016937895, 8.608062959824889940772216486756, 9.762639159623929431209071468560
