| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 11-s + 12-s + 13-s − 16-s − 4·17-s − 18-s − 8·19-s − 22-s − 3·24-s − 5·25-s − 26-s − 27-s + 4·29-s − 6·31-s − 5·32-s − 33-s + 4·34-s − 36-s − 6·37-s + 8·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.83·19-s − 0.213·22-s − 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.742·29-s − 1.07·31-s − 0.883·32-s − 0.174·33-s + 0.685·34-s − 1/6·36-s − 0.986·37-s + 1.29·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−19.44353012295296, −18.75271763573406, −17.86871299465849, −17.46462513185862, −16.74021811711282, −15.94753031987500, −15.01349755365290, −14.03777792898784, −13.16727004647310, −12.51336226771828, −11.29252528487813, −10.68195930442135, −9.805924573894417, −8.882281902623244, −8.198333501616002, −7.006668413057594, −6.077446718710783, −4.786795566950740, −3.939910762033627, −1.825715196240894, 0,
1.825715196240894, 3.939910762033627, 4.786795566950740, 6.077446718710783, 7.006668413057594, 8.198333501616002, 8.882281902623244, 9.805924573894417, 10.68195930442135, 11.29252528487813, 12.51336226771828, 13.16727004647310, 14.03777792898784, 15.01349755365290, 15.94753031987500, 16.74021811711282, 17.46462513185862, 17.86871299465849, 18.75271763573406, 19.44353012295296
