| L(s) = 1 | + 3.12·3-s + 5-s − 5.12·7-s + 6.77·9-s − 5.69·11-s + 5.24·13-s + 3.12·15-s − 2.70·17-s + 7.52·19-s − 16.0·21-s + 0.00236·23-s + 25-s + 11.7·27-s + 2.52·29-s + 5.08·31-s − 17.7·33-s − 5.12·35-s − 5.87·37-s + 16.4·39-s + 9.60·41-s + 0.949·43-s + 6.77·45-s − 1.93·47-s + 19.3·49-s − 8.46·51-s − 2.82·53-s − 5.69·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 0.447·5-s − 1.93·7-s + 2.25·9-s − 1.71·11-s + 1.45·13-s + 0.807·15-s − 0.656·17-s + 1.72·19-s − 3.49·21-s + 0.000493·23-s + 0.200·25-s + 2.26·27-s + 0.468·29-s + 0.912·31-s − 3.09·33-s − 0.866·35-s − 0.965·37-s + 2.62·39-s + 1.49·41-s + 0.144·43-s + 1.00·45-s − 0.282·47-s + 2.75·49-s − 1.18·51-s − 0.388·53-s − 0.767·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.553410474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.553410474\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 - 0.00236T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 - 5.08T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 - 9.60T + 41T^{2} \) |
| 43 | \( 1 - 0.949T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 3.83T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 + 2.27T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−8.241860041128438126002969907735, −7.42984489805531310875652608869, −6.83437698246370545842728444175, −6.06451062351354138182845895684, −5.24124447117685400713553362881, −4.05046668224936084081790184187, −3.32892442179311608399711596157, −2.88778526355426535767161310910, −2.27857504471690476352536691791, −0.901681194172705437808122595385,
0.901681194172705437808122595385, 2.27857504471690476352536691791, 2.88778526355426535767161310910, 3.32892442179311608399711596157, 4.05046668224936084081790184187, 5.24124447117685400713553362881, 6.06451062351354138182845895684, 6.83437698246370545842728444175, 7.42984489805531310875652608869, 8.241860041128438126002969907735
