L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s − 4·7-s + 3·8-s + 9-s − 2·10-s − 11-s + 12-s + 4·14-s − 2·15-s − 16-s − 2·17-s − 18-s − 2·20-s + 4·21-s + 22-s + 8·23-s − 3·24-s − 25-s − 27-s + 4·28-s − 6·29-s + 2·30-s + 8·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.872·21-s + 0.213·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s − 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.365·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82831259030341079249655300429, −6.87625566807413692768701148012, −6.57112145512025234345666775516, −5.59092668167682561675172765450, −5.10830147517302686718916077334, −4.11432405402007502173846810332, −3.20967362704724919819221481375, −2.18490783609343058155048362276, −1.01190903051797145927138124185, 0,
1.01190903051797145927138124185, 2.18490783609343058155048362276, 3.20967362704724919819221481375, 4.11432405402007502173846810332, 5.10830147517302686718916077334, 5.59092668167682561675172765450, 6.57112145512025234345666775516, 6.87625566807413692768701148012, 7.82831259030341079249655300429