| L(s) = 1 | − 2-s + 4-s − 5-s − 1.65·7-s − 8-s + 10-s − 11-s − 0.395·13-s + 1.65·14-s + 16-s + 2.65·17-s − 6.10·19-s − 20-s + 22-s − 2·23-s + 25-s + 0.395·26-s − 1.65·28-s + 3.70·29-s + 6.49·31-s − 32-s − 2.65·34-s + 1.65·35-s + 4.18·37-s + 6.10·38-s + 40-s + 7.44·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.625·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.109·13-s + 0.442·14-s + 0.250·16-s + 0.644·17-s − 1.40·19-s − 0.223·20-s + 0.213·22-s − 0.417·23-s + 0.200·25-s + 0.0776·26-s − 0.312·28-s + 0.688·29-s + 1.16·31-s − 0.176·32-s − 0.455·34-s + 0.279·35-s + 0.688·37-s + 0.989·38-s + 0.158·40-s + 1.16·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 0.395T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 + 6.10T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 - 7.44T + 41T^{2} \) |
| 43 | \( 1 - 5.44T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.49T + 53T^{2} \) |
| 59 | \( 1 + 2.13T + 59T^{2} \) |
| 61 | \( 1 + 7.39T + 61T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 4.51T + 79T^{2} \) |
| 83 | \( 1 + 0.740T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 3.10T + 97T^{2} \) |
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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
−7.77504394254241380196225947955, −7.20921624272096643887847403847, −6.24310554555886262543313435149, −5.96708807627148067641538510926, −4.69607395873485449533107772846, −4.04794196198712653507184866306, −3.00264689768603550684349934354, −2.38982757522936635529038663461, −1.08714055360800822360616279364, 0,
1.08714055360800822360616279364, 2.38982757522936635529038663461, 3.00264689768603550684349934354, 4.04794196198712653507184866306, 4.69607395873485449533107772846, 5.96708807627148067641538510926, 6.24310554555886262543313435149, 7.20921624272096643887847403847, 7.77504394254241380196225947955
