| L(s) = 1 | − 1.78·3-s − 0.157·5-s + 0.200·9-s − 3.73·11-s − 6.06·13-s + 0.281·15-s + 7.10·17-s − 4.70·19-s − 23-s − 4.97·25-s + 5.00·27-s − 5.99·29-s − 1.31·31-s + 6.68·33-s − 3.66·37-s + 10.8·39-s − 7.95·41-s + 11.1·43-s − 0.0315·45-s + 0.0837·47-s − 12.7·51-s − 9.65·53-s + 0.587·55-s + 8.42·57-s − 10.6·59-s − 1.59·61-s + 0.952·65-s + ⋯ |
| L(s) = 1 | − 1.03·3-s − 0.0702·5-s + 0.0669·9-s − 1.12·11-s − 1.68·13-s + 0.0725·15-s + 1.72·17-s − 1.07·19-s − 0.208·23-s − 0.995·25-s + 0.963·27-s − 1.11·29-s − 0.236·31-s + 1.16·33-s − 0.602·37-s + 1.73·39-s − 1.24·41-s + 1.70·43-s − 0.00470·45-s + 0.0122·47-s − 1.78·51-s − 1.32·53-s + 0.0792·55-s + 1.11·57-s − 1.39·59-s − 0.204·61-s + 0.118·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2339331370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2339331370\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 0.157T + 5T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.31T + 31T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 0.0837T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 1.59T + 61T^{2} \) |
| 67 | \( 1 + 5.89T + 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 3.55T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.60000966395652377448943549410, −7.20630579517893570746130689308, −6.08350556552246745631460375076, −5.71993134620343614174622284446, −5.05805344769137565466065796379, −4.52043118086814466717053041661, −3.41588377269174113248135850372, −2.61891539661228041033910938377, −1.70321144694533958175989599420, −0.23580906989343918026962216958,
0.23580906989343918026962216958, 1.70321144694533958175989599420, 2.61891539661228041033910938377, 3.41588377269174113248135850372, 4.52043118086814466717053041661, 5.05805344769137565466065796379, 5.71993134620343614174622284446, 6.08350556552246745631460375076, 7.20630579517893570746130689308, 7.60000966395652377448943549410
