| L(s) = 1 | − 2.61·2-s − 1.14·4-s − 18.9·5-s + 25.2·7-s + 23.9·8-s + 49.6·10-s − 44.2·11-s − 41.4·13-s − 66.2·14-s − 53.5·16-s − 94.3·17-s + 34.2·19-s + 21.6·20-s + 115.·22-s + 31.8·23-s + 234.·25-s + 108.·26-s − 28.8·28-s + 74.2·29-s − 114.·31-s − 51.2·32-s + 247.·34-s − 479.·35-s + 18.9·37-s − 89.8·38-s − 453.·40-s − 21.2·41-s + ⋯ |
| L(s) = 1 | − 0.925·2-s − 0.142·4-s − 1.69·5-s + 1.36·7-s + 1.05·8-s + 1.57·10-s − 1.21·11-s − 0.884·13-s − 1.26·14-s − 0.837·16-s − 1.34·17-s + 0.414·19-s + 0.241·20-s + 1.12·22-s + 0.288·23-s + 1.87·25-s + 0.819·26-s − 0.194·28-s + 0.475·29-s − 0.664·31-s − 0.282·32-s + 1.24·34-s − 2.31·35-s + 0.0839·37-s − 0.383·38-s − 1.79·40-s − 0.0810·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1911878842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1911878842\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.61T + 8T^{2} \) |
| 5 | \( 1 + 18.9T + 125T^{2} \) |
| 7 | \( 1 - 25.2T + 343T^{2} \) |
| 11 | \( 1 + 44.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 41.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 74.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 18.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 21.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 424.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 106.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 97.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 556.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 244.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 648.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 954.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 132.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 358.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−8.446362640253623006848104989170, −8.074129209645632957292273433250, −7.53044013145571203677986437788, −6.92287108785841686770884955952, −5.07843558039188249288622132428, −4.80370775244310581082721857395, −4.00337140085302271482567398545, −2.71135708404735253358917195969, −1.52316053296817501355106447568, −0.23589485110160262633323022498,
0.23589485110160262633323022498, 1.52316053296817501355106447568, 2.71135708404735253358917195969, 4.00337140085302271482567398545, 4.80370775244310581082721857395, 5.07843558039188249288622132428, 6.92287108785841686770884955952, 7.53044013145571203677986437788, 8.074129209645632957292273433250, 8.446362640253623006848104989170
