| L(s) = 1 | − 2-s + 4-s + 0.585·5-s − 7-s − 8-s − 0.585·10-s + 4.24·11-s − 4.82·13-s + 14-s + 16-s − 4.82·17-s + 7.07·19-s + 0.585·20-s − 4.24·22-s − 23-s − 4.65·25-s + 4.82·26-s − 28-s − 2·29-s − 4.82·31-s − 32-s + 4.82·34-s − 0.585·35-s − 1.75·37-s − 7.07·38-s − 0.585·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.261·5-s − 0.377·7-s − 0.353·8-s − 0.185·10-s + 1.27·11-s − 1.33·13-s + 0.267·14-s + 0.250·16-s − 1.17·17-s + 1.62·19-s + 0.130·20-s − 0.904·22-s − 0.208·23-s − 0.931·25-s + 0.946·26-s − 0.188·28-s − 0.371·29-s − 0.867·31-s − 0.176·32-s + 0.828·34-s − 0.0990·35-s − 0.288·37-s − 1.14·38-s − 0.0926·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 9.41T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 9.41T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.591848881486271745417349589293, −7.37595880480208564535786438365, −7.19415349337987942506842278609, −6.21484594757538370530095672864, −5.47762335478346824339738752843, −4.40993581204849889180077964147, −3.45488949123964462682137576814, −2.41046832405054972692471863907, −1.46441856478860535879544349632, 0,
1.46441856478860535879544349632, 2.41046832405054972692471863907, 3.45488949123964462682137576814, 4.40993581204849889180077964147, 5.47762335478346824339738752843, 6.21484594757538370530095672864, 7.19415349337987942506842278609, 7.37595880480208564535786438365, 8.591848881486271745417349589293
