| L(s) = 1 | − 0.614·2-s − 3-s − 1.62·4-s + 5-s + 0.614·6-s + 2.22·8-s + 9-s − 0.614·10-s − 11-s + 1.62·12-s + 3.55·13-s − 15-s + 1.87·16-s + 5.24·17-s − 0.614·18-s − 7.06·19-s − 1.62·20-s + 0.614·22-s − 4.40·23-s − 2.22·24-s + 25-s − 2.18·26-s − 27-s − 3.14·29-s + 0.614·30-s + 5.88·31-s − 5.60·32-s + ⋯ |
| L(s) = 1 | − 0.434·2-s − 0.577·3-s − 0.811·4-s + 0.447·5-s + 0.250·6-s + 0.787·8-s + 0.333·9-s − 0.194·10-s − 0.301·11-s + 0.468·12-s + 0.986·13-s − 0.258·15-s + 0.469·16-s + 1.27·17-s − 0.144·18-s − 1.62·19-s − 0.362·20-s + 0.131·22-s − 0.918·23-s − 0.454·24-s + 0.200·25-s − 0.428·26-s − 0.192·27-s − 0.583·29-s + 0.112·30-s + 1.05·31-s − 0.990·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.614T + 2T^{2} \) |
| 13 | \( 1 - 3.55T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + 7.06T + 19T^{2} \) |
| 23 | \( 1 + 4.40T + 23T^{2} \) |
| 29 | \( 1 + 3.14T + 29T^{2} \) |
| 31 | \( 1 - 5.88T + 31T^{2} \) |
| 37 | \( 1 + 3.89T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 - 9.22T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 + 7.94T + 79T^{2} \) |
| 83 | \( 1 + 9.86T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.69310624687901353375944448761, −6.70405123219836174618579557117, −5.94545187190316630360449720671, −5.58549684830961588365731561865, −4.60205446254613814196954503929, −4.09738613334908956483770646101, −3.17851950136389733826860051863, −1.91694021837379836527835345054, −1.10494024336358876333872253500, 0,
1.10494024336358876333872253500, 1.91694021837379836527835345054, 3.17851950136389733826860051863, 4.09738613334908956483770646101, 4.60205446254613814196954503929, 5.58549684830961588365731561865, 5.94545187190316630360449720671, 6.70405123219836174618579557117, 7.69310624687901353375944448761
