| L(s) = 1 | + 2·2-s + 9.18·3-s + 4·4-s + 18.3·6-s + 8·8-s + 57.4·9-s + 35.4·11-s + 36.7·12-s − 45.5·13-s + 16·16-s + 93.7·17-s + 114.·18-s + 44.8·19-s + 70.9·22-s − 122.·23-s + 73.5·24-s − 91.1·26-s + 279.·27-s − 17.8·29-s + 27.1·31-s + 32·32-s + 325.·33-s + 187.·34-s + 229.·36-s − 78.0·37-s + 89.7·38-s − 418.·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.76·3-s + 0.5·4-s + 1.25·6-s + 0.353·8-s + 2.12·9-s + 0.972·11-s + 0.884·12-s − 0.971·13-s + 0.250·16-s + 1.33·17-s + 1.50·18-s + 0.541·19-s + 0.687·22-s − 1.11·23-s + 0.625·24-s − 0.687·26-s + 1.99·27-s − 0.114·29-s + 0.157·31-s + 0.176·32-s + 1.71·33-s + 0.945·34-s + 1.06·36-s − 0.346·37-s + 0.383·38-s − 1.71·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.475731921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.475731921\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.18T + 27T^{2} \) |
| 11 | \( 1 - 35.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 27.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 21.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 578.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 161.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 559.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 407.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 256.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 853.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 828.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 164.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 38.6T + 9.12e5T^{2} \) |
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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
−8.541400043610133173875864614617, −7.61141792130024962985769174951, −7.42153985851591542823135620827, −6.32311396790876058545482246651, −5.33112413051734286943172513558, −4.24694863293656160515443609220, −3.71231035322055388268282879997, −2.89313410160646831490469113537, −2.13098962937589573020791056861, −1.14633311604110380382612515697,
1.14633311604110380382612515697, 2.13098962937589573020791056861, 2.89313410160646831490469113537, 3.71231035322055388268282879997, 4.24694863293656160515443609220, 5.33112413051734286943172513558, 6.32311396790876058545482246651, 7.42153985851591542823135620827, 7.61141792130024962985769174951, 8.541400043610133173875864614617
