| L(s) = 1 | − 3.77·2-s + 0.853·3-s + 6.21·4-s − 1.45·5-s − 3.21·6-s + 6.72·8-s − 26.2·9-s + 5.50·10-s + 47.6·11-s + 5.30·12-s + 57.4·13-s − 1.24·15-s − 75.0·16-s + 123.·17-s + 99.0·18-s − 139.·19-s − 9.07·20-s − 179.·22-s + 64.5·23-s + 5.74·24-s − 122.·25-s − 216.·26-s − 45.4·27-s + 217.·29-s + 4.69·30-s + 25.7·31-s + 229.·32-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.164·3-s + 0.776·4-s − 0.130·5-s − 0.219·6-s + 0.297·8-s − 0.972·9-s + 0.174·10-s + 1.30·11-s + 0.127·12-s + 1.22·13-s − 0.0214·15-s − 1.17·16-s + 1.75·17-s + 1.29·18-s − 1.68·19-s − 0.101·20-s − 1.73·22-s + 0.585·23-s + 0.0488·24-s − 0.982·25-s − 1.63·26-s − 0.324·27-s + 1.39·29-s + 0.0285·30-s + 0.149·31-s + 1.26·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.258702325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258702325\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 3.77T + 8T^{2} \) |
| 3 | \( 1 - 0.853T + 27T^{2} \) |
| 5 | \( 1 + 1.45T + 125T^{2} \) |
| 11 | \( 1 - 47.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 25.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 232.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 458.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 211.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 99.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 69.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 81.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 221.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 171.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 111.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 467.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 860.T + 9.12e5T^{2} \) |
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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
−8.797460001636426248363369210843, −8.082655510146830511300136159997, −7.48647847162463589450634293665, −6.28460119479997486801311697128, −5.98100860577186945472912817847, −4.48189018821071443231670131548, −3.70671305571263209674583932192, −2.58829791838803759265028845768, −1.37435419756804776078961857199, −0.69061825925864474437446054697,
0.69061825925864474437446054697, 1.37435419756804776078961857199, 2.58829791838803759265028845768, 3.70671305571263209674583932192, 4.48189018821071443231670131548, 5.98100860577186945472912817847, 6.28460119479997486801311697128, 7.48647847162463589450634293665, 8.082655510146830511300136159997, 8.797460001636426248363369210843
