| L(s) = 1 | − 2-s + 4-s − 2.47·5-s − 8-s + 2.47·10-s − 11-s − 3.03·13-s + 16-s − 6.64·17-s − 0.557·19-s − 2.47·20-s + 22-s + 2.66·23-s + 1.11·25-s + 3.03·26-s − 3.77·29-s − 2.00·31-s − 32-s + 6.64·34-s − 0.158·37-s + 0.557·38-s + 2.47·40-s − 5.93·41-s + 4.32·43-s − 44-s − 2.66·46-s − 7.38·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.10·5-s − 0.353·8-s + 0.782·10-s − 0.301·11-s − 0.840·13-s + 0.250·16-s − 1.61·17-s − 0.127·19-s − 0.553·20-s + 0.213·22-s + 0.556·23-s + 0.223·25-s + 0.594·26-s − 0.701·29-s − 0.360·31-s − 0.176·32-s + 1.13·34-s − 0.0260·37-s + 0.0904·38-s + 0.391·40-s − 0.926·41-s + 0.659·43-s − 0.150·44-s − 0.393·46-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2655469127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2655469127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2.47T + 5T^{2} \) |
| 13 | \( 1 + 3.03T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 0.557T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 + 2.00T + 31T^{2} \) |
| 37 | \( 1 + 0.158T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 7.38T + 47T^{2} \) |
| 53 | \( 1 + 2.83T + 53T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.18T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 - 4.59T + 83T^{2} \) |
| 89 | \( 1 + 6.62T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
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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
−7.79039412037028850243547758288, −7.06062059584348111931958548131, −6.68454492125963656736192929203, −5.67693596154000315968146650963, −4.80905643267898948794957069397, −4.21995658011911024667627159144, −3.33641082057976988580846083004, −2.53190155302618950413180578835, −1.67065542604690892796760911574, −0.26329443169943836493114912312,
0.26329443169943836493114912312, 1.67065542604690892796760911574, 2.53190155302618950413180578835, 3.33641082057976988580846083004, 4.21995658011911024667627159144, 4.80905643267898948794957069397, 5.67693596154000315968146650963, 6.68454492125963656736192929203, 7.06062059584348111931958548131, 7.79039412037028850243547758288
