| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4·11-s − 12-s − 2·13-s + 16-s − 17-s − 18-s − 4·19-s − 4·22-s − 4·23-s + 24-s + 2·26-s − 27-s + 2·29-s − 4·31-s − 32-s − 4·33-s + 34-s + 36-s + 6·37-s + 4·38-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.176·32-s − 0.696·33-s + 0.171·34-s + 1/6·36-s + 0.986·37-s + 0.648·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.508076357926522304587707936931, −7.85391852680015313417178593889, −6.81293701478766628611331020079, −6.47749789975439920149959740273, −5.55607683757389846715119248044, −4.51444126045046681585118943777, −3.71442293834107121729389823605, −2.36951231460413935519804036920, −1.37551986425423722963512172703, 0,
1.37551986425423722963512172703, 2.36951231460413935519804036920, 3.71442293834107121729389823605, 4.51444126045046681585118943777, 5.55607683757389846715119248044, 6.47749789975439920149959740273, 6.81293701478766628611331020079, 7.85391852680015313417178593889, 8.508076357926522304587707936931
