| L(s) = 1 | − 0.618·2-s + 3-s − 1.61·4-s − 0.618·6-s + 2·7-s + 2.23·8-s + 9-s − 5.23·11-s − 1.61·12-s + 13-s − 1.23·14-s + 1.85·16-s + 3.47·17-s − 0.618·18-s − 1.76·19-s + 2·21-s + 3.23·22-s − 2.47·23-s + 2.23·24-s − 0.618·26-s + 27-s − 3.23·28-s + 1.23·29-s − 6·31-s − 5.61·32-s − 5.23·33-s − 2.14·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.577·3-s − 0.809·4-s − 0.252·6-s + 0.755·7-s + 0.790·8-s + 0.333·9-s − 1.57·11-s − 0.467·12-s + 0.277·13-s − 0.330·14-s + 0.463·16-s + 0.842·17-s − 0.145·18-s − 0.404·19-s + 0.436·21-s + 0.689·22-s − 0.515·23-s + 0.456·24-s − 0.121·26-s + 0.192·27-s − 0.611·28-s + 0.229·29-s − 1.07·31-s − 0.993·32-s − 0.911·33-s − 0.368·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 5.23T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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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
−7.82490581363402609895073930120, −7.18195783210622157068729718330, −5.93214137300887248246665424039, −5.30541521328240020489861122709, −4.66567364223158795447316061476, −3.94508790956524740744544611983, −3.07306552296843957228983013098, −2.14224589068375308782464962675, −1.23681592256925722731025980959, 0,
1.23681592256925722731025980959, 2.14224589068375308782464962675, 3.07306552296843957228983013098, 3.94508790956524740744544611983, 4.66567364223158795447316061476, 5.30541521328240020489861122709, 5.93214137300887248246665424039, 7.18195783210622157068729718330, 7.82490581363402609895073930120
