| L(s) = 1 | − 1.08·2-s + 23.1·3-s − 30.8·4-s − 25·5-s − 25.0·6-s − 27.0·7-s + 68.0·8-s + 293.·9-s + 27.0·10-s − 121·11-s − 714.·12-s − 478.·13-s + 29.2·14-s − 579.·15-s + 912.·16-s + 225.·17-s − 317.·18-s − 361·19-s + 770.·20-s − 625.·21-s + 131.·22-s + 1.47e3·23-s + 1.57e3·24-s + 625·25-s + 518.·26-s + 1.17e3·27-s + 832.·28-s + ⋯ |
| L(s) = 1 | − 0.191·2-s + 1.48·3-s − 0.963·4-s − 0.447·5-s − 0.284·6-s − 0.208·7-s + 0.375·8-s + 1.20·9-s + 0.0855·10-s − 0.301·11-s − 1.43·12-s − 0.785·13-s + 0.0398·14-s − 0.664·15-s + 0.891·16-s + 0.189·17-s − 0.231·18-s − 0.229·19-s + 0.430·20-s − 0.309·21-s + 0.0577·22-s + 0.583·23-s + 0.558·24-s + 0.200·25-s + 0.150·26-s + 0.309·27-s + 0.200·28-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 | \( 1 + 1.08T + 32T^{2} \) |
| 3 | \( 1 - 23.1T + 243T^{2} \) |
| 7 | \( 1 + 27.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 478.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 225.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.30e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.94e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.31e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.85e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.33e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 134.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.20e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.60e5T + 8.58e9T^{2} \) |
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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
−8.679837391359619385078997872885, −8.070053850604030149328616104509, −7.59428456328774908270579986737, −6.40329744522590498318963457981, −4.93237557974344914908885180752, −4.32725970609078920868833922070, −3.25899405128445056375995782489, −2.62214233570953416529410248808, −1.21005167255474138397228959458, 0,
1.21005167255474138397228959458, 2.62214233570953416529410248808, 3.25899405128445056375995782489, 4.32725970609078920868833922070, 4.93237557974344914908885180752, 6.40329744522590498318963457981, 7.59428456328774908270579986737, 8.070053850604030149328616104509, 8.679837391359619385078997872885
