| L(s) = 1 | + 1.46·2-s − 3·3-s − 5.85·4-s − 9.85·5-s − 4.39·6-s − 29.9·7-s − 20.3·8-s + 9·9-s − 14.4·10-s − 46.9·11-s + 17.5·12-s − 43.8·14-s + 29.5·15-s + 17.0·16-s − 48.2·17-s + 13.1·18-s − 120.·19-s + 57.6·20-s + 89.8·21-s − 68.7·22-s + 130.·23-s + 60.9·24-s − 27.9·25-s − 27·27-s + 175.·28-s − 194.·29-s + 43.3·30-s + ⋯ |
| L(s) = 1 | + 0.518·2-s − 0.577·3-s − 0.731·4-s − 0.881·5-s − 0.299·6-s − 1.61·7-s − 0.897·8-s + 0.333·9-s − 0.456·10-s − 1.28·11-s + 0.422·12-s − 0.837·14-s + 0.508·15-s + 0.266·16-s − 0.688·17-s + 0.172·18-s − 1.45·19-s + 0.644·20-s + 0.933·21-s − 0.666·22-s + 1.18·23-s + 0.518·24-s − 0.223·25-s − 0.192·27-s + 1.18·28-s − 1.24·29-s + 0.263·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08633059665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08633059665\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.46T + 8T^{2} \) |
| 5 | \( 1 + 9.85T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 48.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 32.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 539.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 98.4T + 2.26e5T^{2} \) |
| 67 | \( 1 - 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 150.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 169.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 214.T + 9.12e5T^{2} \) |
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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
−10.57765531400680051875405992926, −9.644397087215426699373850782215, −8.803081485188870075150323220086, −7.71338999698222681205696026017, −6.62074205454994295777568359573, −5.78805966745361168448709319353, −4.70461504707280768006222646963, −3.83020799620250816053407238478, −2.81803957325702872135780170184, −0.16157998886395888825818635129,
0.16157998886395888825818635129, 2.81803957325702872135780170184, 3.83020799620250816053407238478, 4.70461504707280768006222646963, 5.78805966745361168448709319353, 6.62074205454994295777568359573, 7.71338999698222681205696026017, 8.803081485188870075150323220086, 9.644397087215426699373850782215, 10.57765531400680051875405992926
