| L(s) = 1 | + 0.183·2-s + 1.47·3-s − 1.96·4-s + 0.269·6-s + 3.26·7-s − 0.726·8-s − 0.833·9-s + 2·11-s − 2.89·12-s + 0.296·13-s + 0.597·14-s + 3.79·16-s + 5.16·17-s − 0.152·18-s + 1.73·19-s + 4.79·21-s + 0.366·22-s − 0.879·23-s − 1.06·24-s + 0.0542·26-s − 5.64·27-s − 6.41·28-s + 5.91·29-s + 6.09·31-s + 2.14·32-s + 2.94·33-s + 0.945·34-s + ⋯ |
| L(s) = 1 | + 0.129·2-s + 0.849·3-s − 0.983·4-s + 0.110·6-s + 1.23·7-s − 0.256·8-s − 0.277·9-s + 0.603·11-s − 0.835·12-s + 0.0822·13-s + 0.159·14-s + 0.949·16-s + 1.25·17-s − 0.0359·18-s + 0.396·19-s + 1.04·21-s + 0.0781·22-s − 0.183·23-s − 0.218·24-s + 0.0106·26-s − 1.08·27-s − 1.21·28-s + 1.09·29-s + 1.09·31-s + 0.379·32-s + 0.512·33-s + 0.162·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.928465890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928465890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 0.183T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.296T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + 0.879T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 59 | \( 1 - 5.93T + 59T^{2} \) |
| 61 | \( 1 - 0.915T + 61T^{2} \) |
| 67 | \( 1 + 6.88T + 67T^{2} \) |
| 71 | \( 1 + 5.96T + 71T^{2} \) |
| 73 | \( 1 + 8.83T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 7.52T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−10.39104585523522180631634378771, −9.623525360049544829739075331080, −8.564619336534627346958696100470, −8.354888738768259797281195510825, −7.37241583446904839972680271485, −5.84021497099992424412133074707, −4.96473431336400820990180149414, −3.98638222645220691928882734529, −2.96213855548454580222314276960, −1.32604388590584069230662337399,
1.32604388590584069230662337399, 2.96213855548454580222314276960, 3.98638222645220691928882734529, 4.96473431336400820990180149414, 5.84021497099992424412133074707, 7.37241583446904839972680271485, 8.354888738768259797281195510825, 8.564619336534627346958696100470, 9.623525360049544829739075331080, 10.39104585523522180631634378771
