| L(s) = 1 | + 2.12·2-s − 3.47·4-s − 30.7·7-s − 24.4·8-s − 50.1·11-s + 15.9·13-s − 65.2·14-s − 24.0·16-s + 105.·17-s − 21.3·19-s − 106.·22-s + 136.·23-s + 33.9·26-s + 106.·28-s + 224.·29-s − 225.·31-s + 144.·32-s + 224.·34-s + 416.·37-s − 45.2·38-s − 76.1·41-s − 31.7·43-s + 174.·44-s + 289.·46-s + 60.8·47-s + 599.·49-s − 55.5·52-s + ⋯ |
| L(s) = 1 | + 0.751·2-s − 0.434·4-s − 1.65·7-s − 1.07·8-s − 1.37·11-s + 0.340·13-s − 1.24·14-s − 0.375·16-s + 1.50·17-s − 0.257·19-s − 1.03·22-s + 1.23·23-s + 0.255·26-s + 0.720·28-s + 1.43·29-s − 1.30·31-s + 0.796·32-s + 1.13·34-s + 1.85·37-s − 0.193·38-s − 0.290·41-s − 0.112·43-s + 0.597·44-s + 0.927·46-s + 0.188·47-s + 1.74·49-s − 0.148·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.418824954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418824954\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.12T + 8T^{2} \) |
| 7 | \( 1 + 30.7T + 343T^{2} \) |
| 11 | \( 1 + 50.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 416.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 76.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 31.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 466.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 95.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 87.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 412.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 248.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 552.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 291.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 198.T + 9.12e5T^{2} \) |
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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
−9.986072351732090829283301485424, −9.412203176294440035645715537641, −8.412739929674706111671305661728, −7.36218483126207473098929166947, −6.22547405904543831979635479420, −5.59337233525919254444864782976, −4.60048211768182198913934116324, −3.30822491891973609950467231870, −2.90782732048414088117121038546, −0.59800648615558948276356389523,
0.59800648615558948276356389523, 2.90782732048414088117121038546, 3.30822491891973609950467231870, 4.60048211768182198913934116324, 5.59337233525919254444864782976, 6.22547405904543831979635479420, 7.36218483126207473098929166947, 8.412739929674706111671305661728, 9.412203176294440035645715537641, 9.986072351732090829283301485424
