| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 2·11-s − 6·13-s − 3·15-s − 17-s + 19-s + 6·23-s + 25-s − 9·27-s + 6·29-s + 2·31-s − 6·33-s + 8·37-s + 18·39-s + 3·41-s − 2·43-s + 6·45-s + 6·47-s + 3·51-s + 9·53-s + 2·55-s − 3·57-s − 59-s − 2·61-s − 6·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s − 1.66·13-s − 0.774·15-s − 0.242·17-s + 0.229·19-s + 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.11·29-s + 0.359·31-s − 1.04·33-s + 1.31·37-s + 2.88·39-s + 0.468·41-s − 0.304·43-s + 0.894·45-s + 0.875·47-s + 0.420·51-s + 1.23·53-s + 0.269·55-s − 0.397·57-s − 0.130·59-s − 0.256·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.93867380338336, −12.49002093419814, −12.25983193400060, −11.72956437094213, −11.37823601203364, −10.90498628486948, −10.36468188484719, −10.05686292363753, −9.526354057809902, −9.175544182822901, −8.520920066008455, −7.732983405634691, −7.295696341447288, −6.813628001749863, −6.503362994300317, −5.953561189641533, −5.393972179360878, −5.076536316999034, −4.442463996430066, −4.304678457233280, −3.268847458908284, −2.586281093710232, −2.099229958776472, −1.067578206096841, −0.8885492013207759, 0,
0.8885492013207759, 1.067578206096841, 2.099229958776472, 2.586281093710232, 3.268847458908284, 4.304678457233280, 4.442463996430066, 5.076536316999034, 5.393972179360878, 5.953561189641533, 6.503362994300317, 6.813628001749863, 7.295696341447288, 7.732983405634691, 8.520920066008455, 9.175544182822901, 9.526354057809902, 10.05686292363753, 10.36468188484719, 10.90498628486948, 11.37823601203364, 11.72956437094213, 12.25983193400060, 12.49002093419814, 12.93867380338336
