| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 7-s + 8-s + 9-s − 4·10-s + 11-s + 12-s + 14-s − 4·15-s + 16-s − 2·17-s + 18-s − 4·20-s + 21-s + 22-s − 23-s + 24-s + 11·25-s + 27-s + 28-s − 10·29-s − 4·30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.894·20-s + 0.218·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.188·28-s − 1.85·29-s − 0.730·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−16.49041026445121, −16.20282782730670, −15.42773002017641, −15.03955911396767, −14.81500350189967, −14.02612926858935, −13.41761533446566, −12.79970637588226, −12.21837319021677, −11.67023195459979, −11.21574938551893, −10.73371000213596, −9.795176549611486, −9.030803705428452, −8.322605015080667, −7.927022232111990, −7.265187462323602, −6.816168966994526, −5.874112445820313, −4.926513734890823, −4.357265705876324, −3.807786384116047, −3.267577671265514, −2.372726633397225, −1.329367774499162, 0,
1.329367774499162, 2.372726633397225, 3.267577671265514, 3.807786384116047, 4.357265705876324, 4.926513734890823, 5.874112445820313, 6.816168966994526, 7.265187462323602, 7.927022232111990, 8.322605015080667, 9.030803705428452, 9.795176549611486, 10.73371000213596, 11.21574938551893, 11.67023195459979, 12.21837319021677, 12.79970637588226, 13.41761533446566, 14.02612926858935, 14.81500350189967, 15.03955911396767, 15.42773002017641, 16.20282782730670, 16.49041026445121
