| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 4·11-s − 6·13-s + 16-s + 2·17-s + 3·18-s − 4·22-s + 6·26-s + 6·29-s − 8·31-s − 32-s − 2·34-s − 3·36-s + 10·37-s − 2·41-s − 4·43-s + 4·44-s + 8·47-s − 6·52-s + 2·53-s − 6·58-s + 8·59-s + 14·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.852·22-s + 1.17·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.16·47-s − 0.832·52-s + 0.274·53-s − 0.787·58-s + 1.04·59-s + 1.79·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059736022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059736022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.947791175446811247051314576506, −8.301913068586907274040785472708, −7.42472757379966149042918376603, −6.82123838368350264589622397869, −5.91548647651224347146838875772, −5.14037714155681614978134696209, −4.03694373686491315845893969603, −2.95779572995708486306206498227, −2.11286590149397096544933120998, −0.71479745199007503910946566518,
0.71479745199007503910946566518, 2.11286590149397096544933120998, 2.95779572995708486306206498227, 4.03694373686491315845893969603, 5.14037714155681614978134696209, 5.91548647651224347146838875772, 6.82123838368350264589622397869, 7.42472757379966149042918376603, 8.301913068586907274040785472708, 8.947791175446811247051314576506
