| L(s) = 1 | − 4.77·2-s + 4.58·3-s + 14.7·4-s − 9.06·5-s − 21.8·6-s + 11.8·7-s − 32.3·8-s − 5.97·9-s + 43.2·10-s + 67.7·12-s + 13·13-s − 56.7·14-s − 41.5·15-s + 36.2·16-s − 15.1·17-s + 28.5·18-s + 32.9·19-s − 134.·20-s + 54.5·21-s + 77.8·23-s − 148.·24-s − 42.7·25-s − 62.0·26-s − 151.·27-s + 175.·28-s + 91.3·29-s + 198.·30-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.882·3-s + 1.84·4-s − 0.810·5-s − 1.48·6-s + 0.642·7-s − 1.43·8-s − 0.221·9-s + 1.36·10-s + 1.63·12-s + 0.277·13-s − 1.08·14-s − 0.715·15-s + 0.567·16-s − 0.216·17-s + 0.373·18-s + 0.397·19-s − 1.49·20-s + 0.566·21-s + 0.705·23-s − 1.26·24-s − 0.342·25-s − 0.468·26-s − 1.07·27-s + 1.18·28-s + 0.584·29-s + 1.20·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 4.77T + 8T^{2} \) |
| 3 | \( 1 - 4.58T + 27T^{2} \) |
| 5 | \( 1 + 9.06T + 125T^{2} \) |
| 7 | \( 1 - 11.8T + 343T^{2} \) |
| 17 | \( 1 + 15.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 50.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 72.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 117.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 324.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 333.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 246.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 484.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 881.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 132.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 448.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 970.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 632.T + 9.12e5T^{2} \) |
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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
−8.430892829430663483275177473190, −8.269801581398021302730832395396, −7.45435909520196647573347236811, −6.81101689859101808821206027214, −5.50895978293137674096617848398, −4.21872760063693696809263154886, −3.15372934901298221149655960824, −2.21314253677159075280349702454, −1.16536661278857270667298042885, 0,
1.16536661278857270667298042885, 2.21314253677159075280349702454, 3.15372934901298221149655960824, 4.21872760063693696809263154886, 5.50895978293137674096617848398, 6.81101689859101808821206027214, 7.45435909520196647573347236811, 8.269801581398021302730832395396, 8.430892829430663483275177473190
