| L(s) = 1 | + 2.22·2-s + 2.93·4-s + 0.179·5-s − 5.01·7-s + 2.06·8-s + 0.399·10-s + 11-s + 2.16·13-s − 11.1·14-s − 1.26·16-s + 4.79·17-s + 6.94·19-s + 0.527·20-s + 2.22·22-s − 1.92·23-s − 4.96·25-s + 4.81·26-s − 14.7·28-s − 4.05·29-s − 6.04·31-s − 6.95·32-s + 10.6·34-s − 0.902·35-s + 10.1·37-s + 15.4·38-s + 0.372·40-s + 8.33·41-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.46·4-s + 0.0804·5-s − 1.89·7-s + 0.731·8-s + 0.126·10-s + 0.301·11-s + 0.601·13-s − 2.97·14-s − 0.317·16-s + 1.16·17-s + 1.59·19-s + 0.117·20-s + 0.473·22-s − 0.402·23-s − 0.993·25-s + 0.943·26-s − 2.77·28-s − 0.752·29-s − 1.08·31-s − 1.22·32-s + 1.82·34-s − 0.152·35-s + 1.66·37-s + 2.50·38-s + 0.0588·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.227130561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.227130561\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 5 | \( 1 - 0.179T + 5T^{2} \) |
| 7 | \( 1 + 5.01T + 7T^{2} \) |
| 13 | \( 1 - 2.16T + 13T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 8.33T + 41T^{2} \) |
| 43 | \( 1 - 7.73T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 2.10T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 5.82T + 89T^{2} \) |
| 97 | \( 1 - 0.569T + 97T^{2} \) |
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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
−7.58300204047914818342041903918, −7.25205455312591785318207002810, −6.28146927451913916469077393365, −5.69461876896391663322598017103, −5.59655479995703850859087116168, −4.09381075124293028614389559689, −3.76877111540494318274423791756, −3.10548757977634791596129238419, −2.37094987162548324568483834129, −0.841998162849370296938883250720,
0.841998162849370296938883250720, 2.37094987162548324568483834129, 3.10548757977634791596129238419, 3.76877111540494318274423791756, 4.09381075124293028614389559689, 5.59655479995703850859087116168, 5.69461876896391663322598017103, 6.28146927451913916469077393365, 7.25205455312591785318207002810, 7.58300204047914818342041903918
