| L(s) = 1 | + 2·2-s − 4.75·3-s + 4·4-s − 5·5-s − 9.50·6-s − 10.3·7-s + 8·8-s − 4.42·9-s − 10·10-s − 11·11-s − 19.0·12-s + 25.6·13-s − 20.7·14-s + 23.7·15-s + 16·16-s − 17·17-s − 8.85·18-s + 33.7·19-s − 20·20-s + 49.2·21-s − 22·22-s + 156.·23-s − 38.0·24-s + 25·25-s + 51.2·26-s + 149.·27-s − 41.4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.914·3-s + 0.5·4-s − 0.447·5-s − 0.646·6-s − 0.559·7-s + 0.353·8-s − 0.164·9-s − 0.316·10-s − 0.301·11-s − 0.457·12-s + 0.546·13-s − 0.395·14-s + 0.408·15-s + 0.250·16-s − 0.242·17-s − 0.115·18-s + 0.407·19-s − 0.223·20-s + 0.511·21-s − 0.213·22-s + 1.41·23-s − 0.323·24-s + 0.200·25-s + 0.386·26-s + 1.06·27-s − 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1870 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 4.75T + 27T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 13 | \( 1 - 25.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 33.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 80.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 253.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 253.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 578.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 779.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 703.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 756.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 898.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 869.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 309.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 771.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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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
−8.444836214908043689788540107244, −7.39861678509535192624808524917, −6.73301543828722361319978940267, −5.94659069293987232559444478083, −5.30139469865999312011961027850, −4.48973415179430688352818949415, −3.46651049455714686873267577177, −2.69991783959220630057853449046, −1.13593608218800549213103286756, 0,
1.13593608218800549213103286756, 2.69991783959220630057853449046, 3.46651049455714686873267577177, 4.48973415179430688352818949415, 5.30139469865999312011961027850, 5.94659069293987232559444478083, 6.73301543828722361319978940267, 7.39861678509535192624808524917, 8.444836214908043689788540107244
