| L(s) = 1 | + 2-s + 1.70·3-s + 4-s + 5-s + 1.70·6-s + 1.07·7-s + 8-s − 0.0783·9-s + 10-s − 6.04·11-s + 1.70·12-s + 0.290·13-s + 1.07·14-s + 1.70·15-s + 16-s − 1.07·17-s − 0.0783·18-s + 5.26·19-s + 20-s + 1.84·21-s − 6.04·22-s + 4.34·23-s + 1.70·24-s + 25-s + 0.290·26-s − 5.26·27-s + 1.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.986·3-s + 0.5·4-s + 0.447·5-s + 0.697·6-s + 0.407·7-s + 0.353·8-s − 0.0261·9-s + 0.316·10-s − 1.82·11-s + 0.493·12-s + 0.0806·13-s + 0.288·14-s + 0.441·15-s + 0.250·16-s − 0.261·17-s − 0.0184·18-s + 1.20·19-s + 0.223·20-s + 0.402·21-s − 1.28·22-s + 0.904·23-s + 0.348·24-s + 0.200·25-s + 0.0570·26-s − 1.01·27-s + 0.203·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.290800256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.290800256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.04T + 11T^{2} \) |
| 13 | \( 1 - 0.290T + 13T^{2} \) |
| 17 | \( 1 + 1.07T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 - 9.31T + 29T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + 1.75T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 4.68T + 71T^{2} \) |
| 73 | \( 1 - 8.18T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 3.84T + 89T^{2} \) |
| 97 | \( 1 - 2.49T + 97T^{2} \) |
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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
−7.55580880758408573045834275158, −7.24950828421604510716571244370, −6.15761360485456725699572804419, −5.46952537736907706346903709957, −5.00128957774152633481394473721, −4.22329245395159891974297754349, −3.12784559907594897283994556138, −2.75640274635620554704189251811, −2.17434319735332508259951121257, −0.939655340382282966799395232452,
0.939655340382282966799395232452, 2.17434319735332508259951121257, 2.75640274635620554704189251811, 3.12784559907594897283994556138, 4.22329245395159891974297754349, 5.00128957774152633481394473721, 5.46952537736907706346903709957, 6.15761360485456725699572804419, 7.24950828421604510716571244370, 7.55580880758408573045834275158
