| L(s) = 1 | + 2.23·2-s + 3-s + 2.98·4-s + 2.23·6-s − 1.03·7-s + 2.20·8-s + 9-s + 6.17·11-s + 2.98·12-s + 0.937·13-s − 2.30·14-s − 1.04·16-s − 6.56·17-s + 2.23·18-s + 5.67·19-s − 1.03·21-s + 13.7·22-s + 1.64·23-s + 2.20·24-s + 2.09·26-s + 27-s − 3.08·28-s + 8.35·29-s + 5.53·31-s − 6.74·32-s + 6.17·33-s − 14.6·34-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 0.577·3-s + 1.49·4-s + 0.911·6-s − 0.389·7-s + 0.781·8-s + 0.333·9-s + 1.86·11-s + 0.862·12-s + 0.260·13-s − 0.615·14-s − 0.260·16-s − 1.59·17-s + 0.526·18-s + 1.30·19-s − 0.225·21-s + 2.94·22-s + 0.343·23-s + 0.451·24-s + 0.410·26-s + 0.192·27-s − 0.582·28-s + 1.55·29-s + 0.993·31-s − 1.19·32-s + 1.07·33-s − 2.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.447151495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.447151495\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 7 | \( 1 + 1.03T + 7T^{2} \) |
| 11 | \( 1 - 6.17T + 11T^{2} \) |
| 13 | \( 1 - 0.937T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 5.67T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 - 0.500T + 53T^{2} \) |
| 59 | \( 1 + 1.19T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 1.46T + 83T^{2} \) |
| 89 | \( 1 + 8.51T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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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
−9.134023466243786244989036656347, −8.589502753416992573644026753880, −7.26931127012519355969430526229, −6.53332033579277077894156467202, −6.18413473174725498223878955835, −4.83728188170894312480323240785, −4.29893795253070812661075414764, −3.43261770948563942196773415411, −2.75532563651551502475747797246, −1.44357581412762530873971807093,
1.44357581412762530873971807093, 2.75532563651551502475747797246, 3.43261770948563942196773415411, 4.29893795253070812661075414764, 4.83728188170894312480323240785, 6.18413473174725498223878955835, 6.53332033579277077894156467202, 7.26931127012519355969430526229, 8.589502753416992573644026753880, 9.134023466243786244989036656347
