L(s) = 1 | + 26.8·2-s + 81·3-s + 210.·4-s + 1.00e3·5-s + 2.17e3·6-s + 2.54e3·7-s − 8.10e3·8-s + 6.56e3·9-s + 2.70e4·10-s − 1.70e4·11-s + 1.70e4·12-s + 1.77e5·13-s + 6.82e4·14-s + 8.13e4·15-s − 3.25e5·16-s − 4.94e4·17-s + 1.76e5·18-s + 5.60e5·19-s + 2.11e5·20-s + 2.05e5·21-s − 4.58e5·22-s − 6.59e5·23-s − 6.56e5·24-s − 9.44e5·25-s + 4.76e6·26-s + 5.31e5·27-s + 5.35e5·28-s + ⋯ |
L(s) = 1 | + 1.18·2-s + 0.577·3-s + 0.411·4-s + 0.718·5-s + 0.685·6-s + 0.399·7-s − 0.699·8-s + 0.333·9-s + 0.853·10-s − 0.351·11-s + 0.237·12-s + 1.72·13-s + 0.475·14-s + 0.414·15-s − 1.24·16-s − 0.143·17-s + 0.396·18-s + 0.987·19-s + 0.295·20-s + 0.230·21-s − 0.417·22-s − 0.491·23-s − 0.403·24-s − 0.483·25-s + 2.04·26-s + 0.192·27-s + 0.164·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(6.333788695\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.333788695\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
good | 2 | \( 1 - 26.8T + 512T^{2} \) |
| 5 | \( 1 - 1.00e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.54e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.70e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.77e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.94e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.60e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.59e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.93e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.19e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.70e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.16e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.46e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.15e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.75e6T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.29e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.48e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.49e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.10e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 6.21e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.85e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.17e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.93e8T + 7.60e17T^{2} \) |
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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
−11.22214660034819007997690343816, −10.01774675369191099171178931244, −8.949492754213263439179528943265, −8.028202861733753515861912320898, −6.45193084644361543314158150446, −5.66038460383325707607575720283, −4.52536595730593628244785151508, −3.48242310796103419342116464684, −2.43271531151244374933583007368, −1.08822027364610436890829653083,
1.08822027364610436890829653083, 2.43271531151244374933583007368, 3.48242310796103419342116464684, 4.52536595730593628244785151508, 5.66038460383325707607575720283, 6.45193084644361543314158150446, 8.028202861733753515861912320898, 8.949492754213263439179528943265, 10.01774675369191099171178931244, 11.22214660034819007997690343816