| L(s) = 1 | + 0.711·2-s + 0.689·3-s − 7.49·4-s + 0.489·6-s + 19.0·7-s − 11.0·8-s − 26.5·9-s + 50.7·11-s − 5.16·12-s − 45.4·13-s + 13.5·14-s + 52.1·16-s − 53.0·17-s − 18.8·18-s + 58.8·19-s + 13.1·21-s + 36.0·22-s + 23·23-s − 7.59·24-s − 32.3·26-s − 36.8·27-s − 142.·28-s − 70.7·29-s − 286.·31-s + 125.·32-s + 34.9·33-s − 37.7·34-s + ⋯ |
| L(s) = 1 | + 0.251·2-s + 0.132·3-s − 0.936·4-s + 0.0333·6-s + 1.02·7-s − 0.486·8-s − 0.982·9-s + 1.38·11-s − 0.124·12-s − 0.969·13-s + 0.258·14-s + 0.814·16-s − 0.756·17-s − 0.246·18-s + 0.710·19-s + 0.136·21-s + 0.349·22-s + 0.208·23-s − 0.0645·24-s − 0.243·26-s − 0.262·27-s − 0.963·28-s − 0.452·29-s − 1.66·31-s + 0.691·32-s + 0.184·33-s − 0.190·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.711T + 8T^{2} \) |
| 3 | \( 1 - 0.689T + 27T^{2} \) |
| 7 | \( 1 - 19.0T + 343T^{2} \) |
| 11 | \( 1 - 50.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 58.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 70.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 286.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 95.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 148.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 379.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 741.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 5.17T + 2.26e5T^{2} \) |
| 67 | \( 1 + 974.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 336.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 317.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 577.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 225.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 147.T + 9.12e5T^{2} \) |
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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
−9.532172738980418731730914842952, −9.094046840498418782973355597197, −8.249014966735985458394064996305, −7.30214661682020467055729248416, −5.99568440001096561679377144935, −5.07366911280769106181196934714, −4.28909766514303131114496830879, −3.15415010887968857558354501397, −1.60672010936864913147097629751, 0,
1.60672010936864913147097629751, 3.15415010887968857558354501397, 4.28909766514303131114496830879, 5.07366911280769106181196934714, 5.99568440001096561679377144935, 7.30214661682020467055729248416, 8.249014966735985458394064996305, 9.094046840498418782973355597197, 9.532172738980418731730914842952
