| L(s) = 1 | − 2.08·2-s + 2.35·4-s − 5.08·7-s − 0.734·8-s + 4.17·11-s + 1.08·13-s + 10.6·14-s − 3.17·16-s + 0.648·17-s − 2.70·19-s − 8.70·22-s + 7.52·23-s − 2.26·26-s − 11.9·28-s − 5.90·29-s + 31-s + 8.08·32-s − 1.35·34-s − 0.913·37-s + 5.64·38-s − 2.17·41-s − 8.17·43-s + 9.81·44-s − 15.6·46-s − 10.2·47-s + 18.8·49-s + 2.55·52-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.17·4-s − 1.92·7-s − 0.259·8-s + 1.25·11-s + 0.301·13-s + 2.83·14-s − 0.793·16-s + 0.157·17-s − 0.620·19-s − 1.85·22-s + 1.56·23-s − 0.444·26-s − 2.26·28-s − 1.09·29-s + 0.179·31-s + 1.42·32-s − 0.231·34-s − 0.150·37-s + 0.915·38-s − 0.339·41-s − 1.24·43-s + 1.47·44-s − 2.31·46-s − 1.49·47-s + 2.69·49-s + 0.354·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 7 | \( 1 + 5.08T + 7T^{2} \) |
| 11 | \( 1 - 4.17T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 29 | \( 1 + 5.90T + 29T^{2} \) |
| 37 | \( 1 + 0.913T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + 0.438T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 2.96T + 71T^{2} \) |
| 73 | \( 1 - 1.25T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.61888062709925484704336895646, −6.82097504266487837799537874792, −6.64192377420199934969948885708, −5.88180989733868710214182075310, −4.69500520995715511114339683155, −3.65337530601741352712503282861, −3.13685605144588961950395001449, −1.97683051332487313314013389890, −0.988147011255202477562530668569, 0,
0.988147011255202477562530668569, 1.97683051332487313314013389890, 3.13685605144588961950395001449, 3.65337530601741352712503282861, 4.69500520995715511114339683155, 5.88180989733868710214182075310, 6.64192377420199934969948885708, 6.82097504266487837799537874792, 7.61888062709925484704336895646
