| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s + 2·13-s − 15-s + 16-s − 6·17-s − 18-s + 19-s + 20-s − 23-s + 24-s + 25-s − 2·26-s − 27-s − 2·29-s + 30-s − 8·31-s − 32-s + 6·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−16.49961064633771, −16.01726182253216, −15.75452352877368, −14.80704631962574, −14.49749473475921, −13.50062547512364, −13.14262568172970, −12.59298825195846, −11.73752115470970, −11.32511570482990, −10.74443462237254, −10.34104519043910, −9.526132080428732, −9.033775108484079, −8.587061171153447, −7.641200983528487, −7.146877428043509, −6.471869893543190, −5.875548388576909, −5.351728643453434, −4.378319747993671, −3.753252865330316, −2.624996094367984, −1.956029714089635, −1.060721511783229, 0,
1.060721511783229, 1.956029714089635, 2.624996094367984, 3.753252865330316, 4.378319747993671, 5.351728643453434, 5.875548388576909, 6.471869893543190, 7.146877428043509, 7.641200983528487, 8.587061171153447, 9.033775108484079, 9.526132080428732, 10.34104519043910, 10.74443462237254, 11.32511570482990, 11.73752115470970, 12.59298825195846, 13.14262568172970, 13.50062547512364, 14.49749473475921, 14.80704631962574, 15.75452352877368, 16.01726182253216, 16.49961064633771
