| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s + 4·13-s + 3·14-s + 15-s + 16-s − 7·17-s + 18-s + 20-s + 3·21-s − 3·22-s − 23-s + 24-s − 4·25-s + 4·26-s + 27-s + 3·28-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 1.10·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.235·18-s + 0.223·20-s + 0.654·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.784·26-s + 0.192·27-s + 0.566·28-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.280672363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.280672363\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.56450799702408532716742528222, −9.373316090770423603669230423025, −8.414634231802839197356182512687, −7.84571695344488828751904204334, −6.70723751373357147176564043735, −5.77032904182661827958621201672, −4.78772749588110161256930225537, −3.97419899782622180724046335734, −2.60690322756399036254963846361, −1.71239652171907282151966562345,
1.71239652171907282151966562345, 2.60690322756399036254963846361, 3.97419899782622180724046335734, 4.78772749588110161256930225537, 5.77032904182661827958621201672, 6.70723751373357147176564043735, 7.84571695344488828751904204334, 8.414634231802839197356182512687, 9.373316090770423603669230423025, 10.56450799702408532716742528222
