| L(s) = 1 | + 2.72·2-s − 2.29·3-s + 5.41·4-s − 5-s − 6.24·6-s + 3.82·7-s + 9.30·8-s + 2.25·9-s − 2.72·10-s − 4.41·11-s − 12.4·12-s − 3.67·13-s + 10.4·14-s + 2.29·15-s + 14.5·16-s − 2.28·17-s + 6.14·18-s − 2.39·19-s − 5.41·20-s − 8.77·21-s − 12.0·22-s − 0.265·23-s − 21.3·24-s + 25-s − 10.0·26-s + 1.70·27-s + 20.7·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 1.32·3-s + 2.70·4-s − 0.447·5-s − 2.54·6-s + 1.44·7-s + 3.29·8-s + 0.752·9-s − 0.861·10-s − 1.33·11-s − 3.58·12-s − 1.01·13-s + 2.78·14-s + 0.592·15-s + 3.62·16-s − 0.554·17-s + 1.44·18-s − 0.548·19-s − 1.21·20-s − 1.91·21-s − 2.56·22-s − 0.0553·23-s − 4.35·24-s + 0.200·25-s − 1.96·26-s + 0.327·27-s + 3.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.308398003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.308398003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 3 | \( 1 + 2.29T + 3T^{2} \) |
| 7 | \( 1 - 3.82T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 + 0.265T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.109T + 53T^{2} \) |
| 59 | \( 1 + 2.00T + 59T^{2} \) |
| 61 | \( 1 - 3.96T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 + 0.140T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−12.46255944100228155947207988645, −11.84267740631140168356801220595, −11.01565153991037924670571636370, −10.60771610289190051814467976226, −7.918525139150551152019140953404, −7.05876834498019419392193677384, −5.71100562208432374945143308111, −5.04539116693120770066126784883, −4.33455911454823304428546661972, −2.34697204956410755155792042680,
2.34697204956410755155792042680, 4.33455911454823304428546661972, 5.04539116693120770066126784883, 5.71100562208432374945143308111, 7.05876834498019419392193677384, 7.918525139150551152019140953404, 10.60771610289190051814467976226, 11.01565153991037924670571636370, 11.84267740631140168356801220595, 12.46255944100228155947207988645
