| L(s) = 1 | − 2·2-s + 3.85·3-s + 4·4-s − 5·5-s − 7.70·6-s + 3.86·7-s − 8·8-s − 12.1·9-s + 10·10-s − 11·11-s + 15.4·12-s + 56.6·13-s − 7.72·14-s − 19.2·15-s + 16·16-s + 17·17-s + 24.3·18-s + 60.4·19-s − 20·20-s + 14.8·21-s + 22·22-s − 118.·23-s − 30.8·24-s + 25·25-s − 113.·26-s − 150.·27-s + 15.4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.741·3-s + 0.5·4-s − 0.447·5-s − 0.524·6-s + 0.208·7-s − 0.353·8-s − 0.450·9-s + 0.316·10-s − 0.301·11-s + 0.370·12-s + 1.20·13-s − 0.147·14-s − 0.331·15-s + 0.250·16-s + 0.242·17-s + 0.318·18-s + 0.729·19-s − 0.223·20-s + 0.154·21-s + 0.213·22-s − 1.07·23-s − 0.262·24-s + 0.200·25-s − 0.854·26-s − 1.07·27-s + 0.104·28-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 | \( 1 - 3.85T + 27T^{2} \) |
| 7 | \( 1 - 3.86T + 343T^{2} \) |
| 13 | \( 1 - 56.6T + 2.19e3T^{2} \) |
| 19 | \( 1 - 60.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 29.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 121.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 390.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 278.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 32.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 23.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 361.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 72.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 500.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 378.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 462.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 77.9T + 9.12e5T^{2} \) |
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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
−8.391873771483650905657054323239, −7.936047318045519203832649430440, −7.26644418856091095164662359299, −6.12149507410629862259488416863, −5.43508733902047434006740418973, −4.01814976285129970572729440703, −3.32630246797553209491978397961, −2.35052630956997585808666970258, −1.26729071320191914598449623710, 0,
1.26729071320191914598449623710, 2.35052630956997585808666970258, 3.32630246797553209491978397961, 4.01814976285129970572729440703, 5.43508733902047434006740418973, 6.12149507410629862259488416863, 7.26644418856091095164662359299, 7.936047318045519203832649430440, 8.391873771483650905657054323239
