| L(s) = 1 | − 9.24·3-s + 59.2·5-s − 65.2·7-s − 157.·9-s − 166.·11-s − 746.·13-s − 547.·15-s + 831.·17-s + 1.04e3·19-s + 603.·21-s + 1.79e3·23-s + 386.·25-s + 3.70e3·27-s + 8.55e3·29-s − 1.55e3·31-s + 1.54e3·33-s − 3.86e3·35-s + 792.·37-s + 6.90e3·39-s + 2.67e3·41-s + 1.61e4·43-s − 9.33e3·45-s − 2.91e4·47-s − 1.25e4·49-s − 7.68e3·51-s − 1.68e4·53-s − 9.88e3·55-s + ⋯ |
| L(s) = 1 | − 0.592·3-s + 1.06·5-s − 0.503·7-s − 0.648·9-s − 0.415·11-s − 1.22·13-s − 0.628·15-s + 0.697·17-s + 0.665·19-s + 0.298·21-s + 0.709·23-s + 0.123·25-s + 0.977·27-s + 1.88·29-s − 0.291·31-s + 0.246·33-s − 0.533·35-s + 0.0952·37-s + 0.726·39-s + 0.248·41-s + 1.33·43-s − 0.687·45-s − 1.92·47-s − 0.746·49-s − 0.413·51-s − 0.822·53-s − 0.440·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 9.24T + 243T^{2} \) |
| 5 | \( 1 - 59.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 65.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 166.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 746.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 831.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.79e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 792.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.67e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.57e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.47e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.16e3T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.954747961850795075847954045341, −7.924550653286835260065532312293, −6.92787357501132541412818450890, −6.13039389472948503644412175338, −5.39433309970864261697101296058, −4.78031211615741518799359088761, −3.12332982152950286478424245891, −2.47469157520665136237319480425, −1.10425752171164358189725916382, 0,
1.10425752171164358189725916382, 2.47469157520665136237319480425, 3.12332982152950286478424245891, 4.78031211615741518799359088761, 5.39433309970864261697101296058, 6.13039389472948503644412175338, 6.92787357501132541412818450890, 7.924550653286835260065532312293, 8.954747961850795075847954045341
