| L(s) = 1 | − 3-s − 3·5-s − 2·9-s + 3·11-s − 2·13-s + 3·15-s + 3·17-s + 19-s + 3·23-s + 4·25-s + 5·27-s + 6·29-s − 7·31-s − 3·33-s + 37-s + 2·39-s + 6·41-s + 4·43-s + 6·45-s − 9·47-s − 3·51-s − 3·53-s − 9·55-s − 57-s − 9·59-s + 61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.904·11-s − 0.554·13-s + 0.774·15-s + 0.727·17-s + 0.229·19-s + 0.625·23-s + 4/5·25-s + 0.962·27-s + 1.11·29-s − 1.25·31-s − 0.522·33-s + 0.164·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.894·45-s − 1.31·47-s − 0.420·51-s − 0.412·53-s − 1.21·55-s − 0.132·57-s − 1.17·59-s + 0.128·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.227904921128916434268724018455, −7.56240616311991526194210715543, −6.87229450800291080599835044327, −6.05313952980566733119414581451, −5.18634468459653321605027216331, −4.42986893621370772138711109774, −3.57954413035331597786593448348, −2.80019703611351490822885633419, −1.16455596775948714060930265899, 0,
1.16455596775948714060930265899, 2.80019703611351490822885633419, 3.57954413035331597786593448348, 4.42986893621370772138711109774, 5.18634468459653321605027216331, 6.05313952980566733119414581451, 6.87229450800291080599835044327, 7.56240616311991526194210715543, 8.227904921128916434268724018455
