| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s + 12-s − 5·13-s + 16-s − 6·17-s − 2·18-s + 5·19-s + 3·23-s + 24-s − 5·26-s − 5·27-s + 10·31-s + 32-s − 6·34-s − 2·36-s − 2·37-s + 5·38-s − 5·39-s + 8·43-s + 3·46-s − 6·47-s + 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.288·12-s − 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.471·18-s + 1.14·19-s + 0.625·23-s + 0.204·24-s − 0.980·26-s − 0.962·27-s + 1.79·31-s + 0.176·32-s − 1.02·34-s − 1/3·36-s − 0.328·37-s + 0.811·38-s − 0.800·39-s + 1.21·43-s + 0.442·46-s − 0.875·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.647382999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647382999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.69045079884779, −12.29470016348095, −11.77250588124772, −11.49845505024537, −10.97348488398108, −10.45307568011586, −9.961864454174526, −9.276300583572241, −9.191426324470156, −8.524788351766169, −7.944593931606423, −7.563709154847908, −7.127956599470799, −6.544030424164674, −6.093951669298352, −5.563168304932275, −4.906881834034953, −4.609506584047632, −4.199530430425163, −3.212550097184507, −3.025155790999811, −2.565882368776782, −1.992875325128388, −1.297735707922814, −0.3472702316907885,
0.3472702316907885, 1.297735707922814, 1.992875325128388, 2.565882368776782, 3.025155790999811, 3.212550097184507, 4.199530430425163, 4.609506584047632, 4.906881834034953, 5.563168304932275, 6.093951669298352, 6.544030424164674, 7.127956599470799, 7.563709154847908, 7.944593931606423, 8.524788351766169, 9.191426324470156, 9.276300583572241, 9.961864454174526, 10.45307568011586, 10.97348488398108, 11.49845505024537, 11.77250588124772, 12.29470016348095, 12.69045079884779
