| L(s) = 1 | + 2-s + 1.44·3-s + 4-s + 1.44·6-s + 3.25·7-s + 8-s − 0.918·9-s + 1.32·11-s + 1.44·12-s − 1.25·13-s + 3.25·14-s + 16-s + 4.62·17-s − 0.918·18-s − 3.37·19-s + 4.70·21-s + 1.32·22-s + 23-s + 1.44·24-s − 1.25·26-s − 5.65·27-s + 3.25·28-s + 4.29·29-s + 2.37·31-s + 32-s + 1.91·33-s + 4.62·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.832·3-s + 0.5·4-s + 0.588·6-s + 1.23·7-s + 0.353·8-s − 0.306·9-s + 0.400·11-s + 0.416·12-s − 0.349·13-s + 0.870·14-s + 0.250·16-s + 1.12·17-s − 0.216·18-s − 0.773·19-s + 1.02·21-s + 0.283·22-s + 0.208·23-s + 0.294·24-s − 0.246·26-s − 1.08·27-s + 0.615·28-s + 0.797·29-s + 0.426·31-s + 0.176·32-s + 0.333·33-s + 0.792·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.677406832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.677406832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 + 1.25T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 29 | \( 1 - 4.29T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 5.13T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 6.06T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 - 0.265T + 71T^{2} \) |
| 73 | \( 1 - 5.26T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 2.02T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.62T + 97T^{2} \) |
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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
−9.804234468807843920293703496491, −8.754734201822628383571527728134, −8.147227672654955273387063620301, −7.46901157178198887534963533059, −6.38081402349667740733228882197, −5.36348333571167038388884162425, −4.59047419049132052252040086370, −3.57587339958905949828239086026, −2.61055359397638914470304847721, −1.54300660971579174399171142752,
1.54300660971579174399171142752, 2.61055359397638914470304847721, 3.57587339958905949828239086026, 4.59047419049132052252040086370, 5.36348333571167038388884162425, 6.38081402349667740733228882197, 7.46901157178198887534963533059, 8.147227672654955273387063620301, 8.754734201822628383571527728134, 9.804234468807843920293703496491
