| L(s) = 1 | − 0.0898·2-s − 1.99·4-s − 2.73·5-s + 4.94·7-s + 0.358·8-s + 0.246·10-s + 0.583·11-s + 1.98·13-s − 0.444·14-s + 3.95·16-s + 0.412·17-s − 19-s + 5.45·20-s − 0.0524·22-s + 5.19·23-s + 2.49·25-s − 0.178·26-s − 9.85·28-s + 1.76·29-s + 7.75·31-s − 1.07·32-s − 0.0370·34-s − 13.5·35-s + 7.04·37-s + 0.0898·38-s − 0.982·40-s − 3.64·41-s + ⋯ |
| L(s) = 1 | − 0.0635·2-s − 0.995·4-s − 1.22·5-s + 1.86·7-s + 0.126·8-s + 0.0778·10-s + 0.175·11-s + 0.549·13-s − 0.118·14-s + 0.987·16-s + 0.100·17-s − 0.229·19-s + 1.21·20-s − 0.0111·22-s + 1.08·23-s + 0.499·25-s − 0.0349·26-s − 1.86·28-s + 0.327·29-s + 1.39·31-s − 0.189·32-s − 0.00635·34-s − 2.28·35-s + 1.15·37-s + 0.0145·38-s − 0.155·40-s − 0.569·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683192972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683192972\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0898T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 11 | \( 1 - 0.583T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 - 0.412T + 17T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 + 3.64T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 0.783T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 8.57T + 67T^{2} \) |
| 71 | \( 1 + 8.22T + 71T^{2} \) |
| 73 | \( 1 + 0.408T + 73T^{2} \) |
| 79 | \( 1 - 9.20T + 79T^{2} \) |
| 83 | \( 1 - 6.84T + 83T^{2} \) |
| 89 | \( 1 + 7.22T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.972249282339828978718366105388, −7.49687761765157704318526628888, −6.50908128031557536845147038497, −5.48680162269838924984497173549, −4.81281588559738268123417391012, −4.36863637572158623231125998262, −3.80499558154322512991706483935, −2.78673980914316851637246676939, −1.43938039203062587206879949163, −0.73938815895793776788802191980,
0.73938815895793776788802191980, 1.43938039203062587206879949163, 2.78673980914316851637246676939, 3.80499558154322512991706483935, 4.36863637572158623231125998262, 4.81281588559738268123417391012, 5.48680162269838924984497173549, 6.50908128031557536845147038497, 7.49687761765157704318526628888, 7.972249282339828978718366105388
