| L(s) = 1 | + 5-s − 1.34·7-s − 4.41·11-s + 2.41·13-s + 1.76·17-s − 2.41·19-s − 23-s + 25-s + 3.76·29-s + 1.34·31-s − 1.34·35-s + 10.8·37-s − 10.6·41-s − 2.83·43-s − 7.80·47-s − 5.18·49-s + 9.29·53-s − 4.41·55-s + 2.46·59-s + 4.41·61-s + 2.41·65-s + 10.8·67-s − 2.23·71-s − 7.95·73-s + 5.95·77-s − 8·79-s − 11.7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.509·7-s − 1.33·11-s + 0.671·13-s + 0.428·17-s − 0.555·19-s − 0.208·23-s + 0.200·25-s + 0.699·29-s + 0.241·31-s − 0.227·35-s + 1.78·37-s − 1.65·41-s − 0.432·43-s − 1.13·47-s − 0.740·49-s + 1.27·53-s − 0.595·55-s + 0.320·59-s + 0.565·61-s + 0.300·65-s + 1.32·67-s − 0.265·71-s − 0.930·73-s + 0.678·77-s − 0.900·79-s − 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 1.34T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 - 9.29T + 53T^{2} \) |
| 59 | \( 1 - 2.46T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 + 7.95T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.48195309284796655990903622386, −6.64499102970027243840604850828, −6.13225864114793256769040514171, −5.40155735676842053468767319183, −4.76706165499783961628323688139, −3.82112164509358145366512216445, −2.98504301941135983848302308491, −2.35202821164459295111660446578, −1.25197726034621787605994447202, 0,
1.25197726034621787605994447202, 2.35202821164459295111660446578, 2.98504301941135983848302308491, 3.82112164509358145366512216445, 4.76706165499783961628323688139, 5.40155735676842053468767319183, 6.13225864114793256769040514171, 6.64499102970027243840604850828, 7.48195309284796655990903622386
