| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 3·5-s + 2·6-s + 7-s + 9-s + 6·10-s − 2·12-s − 2·13-s − 2·14-s + 3·15-s − 4·16-s − 3·17-s − 2·18-s + 4·19-s − 6·20-s − 21-s + 2·23-s + 4·25-s + 4·26-s − 27-s + 2·28-s − 8·29-s − 6·30-s + 2·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s + 1.89·10-s − 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.774·15-s − 16-s − 0.727·17-s − 0.471·18-s + 0.917·19-s − 1.34·20-s − 0.218·21-s + 0.417·23-s + 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 1.48·29-s − 1.09·30-s + 0.359·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.339456951965670048577461229072, −7.907783186490500721305384274271, −7.24285510392732021642851642045, −6.66712333788914099144608039670, −5.33624238861814784550153512921, −4.55637434169763059384885808488, −3.70178855230288849642967320144, −2.30761807062046411026871322881, −1.01218526035501045171800683592, 0,
1.01218526035501045171800683592, 2.30761807062046411026871322881, 3.70178855230288849642967320144, 4.55637434169763059384885808488, 5.33624238861814784550153512921, 6.66712333788914099144608039670, 7.24285510392732021642851642045, 7.907783186490500721305384274271, 8.339456951965670048577461229072
