| L(s) = 1 | − 1.44·2-s − 2.79·3-s + 0.0740·4-s + 2.61·5-s + 4.02·6-s + 4.99·7-s + 2.77·8-s + 4.79·9-s − 3.76·10-s + 0.0546·11-s − 0.206·12-s − 7.19·14-s − 7.30·15-s − 4.14·16-s + 5.08·17-s − 6.90·18-s − 1.17·19-s + 0.193·20-s − 13.9·21-s − 0.0786·22-s + 7.81·23-s − 7.74·24-s + 1.85·25-s − 5.00·27-s + 0.369·28-s − 7.02·29-s + 10.5·30-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 1.61·3-s + 0.0370·4-s + 1.17·5-s + 1.64·6-s + 1.88·7-s + 0.980·8-s + 1.59·9-s − 1.19·10-s + 0.0164·11-s − 0.0596·12-s − 1.92·14-s − 1.88·15-s − 1.03·16-s + 1.23·17-s − 1.62·18-s − 0.269·19-s + 0.0433·20-s − 3.04·21-s − 0.0167·22-s + 1.62·23-s − 1.58·24-s + 0.370·25-s − 0.962·27-s + 0.0699·28-s − 1.30·29-s + 1.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122797101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122797101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 7 | \( 1 - 4.99T + 7T^{2} \) |
| 11 | \( 1 - 0.0546T + 11T^{2} \) |
| 17 | \( 1 - 5.08T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 - 1.55T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 0.191T + 47T^{2} \) |
| 53 | \( 1 - 8.59T + 53T^{2} \) |
| 59 | \( 1 + 2.91T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 + 1.50T + 67T^{2} \) |
| 71 | \( 1 + 1.44T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + 1.74T + 83T^{2} \) |
| 89 | \( 1 - 2.35T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.136090534168351860282974624900, −7.55330015550050522483533594871, −6.87995899768991989270229190514, −5.82756113919403235099584599227, −5.34402889928456025966907115550, −4.92202695960499822845554382563, −4.07544185793989711778263665738, −2.23412184589813250976070612289, −1.36832390827685821827680628385, −0.872870988605352572557599344350,
0.872870988605352572557599344350, 1.36832390827685821827680628385, 2.23412184589813250976070612289, 4.07544185793989711778263665738, 4.92202695960499822845554382563, 5.34402889928456025966907115550, 5.82756113919403235099584599227, 6.87995899768991989270229190514, 7.55330015550050522483533594871, 8.136090534168351860282974624900
