| L(s) = 1 | − 6.88·2-s + 20.3·3-s + 15.4·4-s − 25·5-s − 140.·6-s + 93.5·7-s + 114.·8-s + 172.·9-s + 172.·10-s − 121·11-s + 313.·12-s + 587.·13-s − 644.·14-s − 509.·15-s − 1.27e3·16-s + 316.·17-s − 1.18e3·18-s − 361·19-s − 385.·20-s + 1.90e3·21-s + 833.·22-s + 815.·23-s + 2.32e3·24-s + 625·25-s − 4.04e3·26-s − 1.44e3·27-s + 1.44e3·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 1.30·3-s + 0.481·4-s − 0.447·5-s − 1.59·6-s + 0.721·7-s + 0.631·8-s + 0.707·9-s + 0.544·10-s − 0.301·11-s + 0.629·12-s + 0.963·13-s − 0.878·14-s − 0.584·15-s − 1.24·16-s + 0.265·17-s − 0.861·18-s − 0.229·19-s − 0.215·20-s + 0.942·21-s + 0.366·22-s + 0.321·23-s + 0.824·24-s + 0.200·25-s − 1.17·26-s − 0.381·27-s + 0.347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 6.88T + 32T^{2} \) |
| 3 | \( 1 - 20.3T + 243T^{2} \) |
| 7 | \( 1 - 93.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 587.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 316.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 815.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 608.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 822.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.61e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 9.51e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.88e4T + 8.58e9T^{2} \) |
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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
−8.668289639803394676458572985885, −8.076539598146347195600875676740, −7.73678030713262269899060249641, −6.69468473033211584649118558375, −5.23143835643484495545673969997, −4.13232343050391309681939226223, −3.26285946025564818546111415583, −2.04926943754760060452282500674, −1.28474537538938308711113553198, 0,
1.28474537538938308711113553198, 2.04926943754760060452282500674, 3.26285946025564818546111415583, 4.13232343050391309681939226223, 5.23143835643484495545673969997, 6.69468473033211584649118558375, 7.73678030713262269899060249641, 8.076539598146347195600875676740, 8.668289639803394676458572985885
