| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 4·11-s + 2·13-s + 16-s + 2·17-s + 4·19-s + 20-s + 4·22-s + 8·23-s + 25-s − 2·26-s − 6·29-s + 8·31-s − 32-s − 2·34-s − 2·37-s − 4·38-s − 40-s + 2·41-s − 12·43-s − 4·44-s − 8·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.312·41-s − 1.82·43-s − 0.603·44-s − 1.17·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.464139424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.464139424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.304251894090548502195997501335, −7.79296437757666025865659030241, −6.98255528939382129016622071465, −6.31377059297515389510341024477, −5.33653720510679489341670431089, −4.96005935131234511702158322186, −3.45775664678991085783713096127, −2.86836128944115569206808585852, −1.79822957267634689842433227663, −0.76963011843977692538064152860,
0.76963011843977692538064152860, 1.79822957267634689842433227663, 2.86836128944115569206808585852, 3.45775664678991085783713096127, 4.96005935131234511702158322186, 5.33653720510679489341670431089, 6.31377059297515389510341024477, 6.98255528939382129016622071465, 7.79296437757666025865659030241, 8.304251894090548502195997501335
