L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 4·11-s − 12-s + 4·13-s + 16-s + 3·17-s + 18-s − 6·19-s − 4·22-s − 7·23-s − 24-s + 4·26-s − 27-s + 4·29-s + 5·31-s + 32-s + 4·33-s + 3·34-s + 36-s − 2·37-s − 6·38-s − 4·39-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 + 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.852·22-s − 1.45·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.742·29-s + 0.898·31-s + 0.176·32-s + 0.696·33-s + 0.514·34-s + 1/6·36-s − 0.328·37-s − 0.973·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−7.47415218255942471976438176842, −6.60220618397810332038818813541, −6.04773962213833599570492747308, −5.54114747838435288152941825677, −4.70321741421989721958914979215, −4.09438452877911047401406976138, −3.24137434564808265370087117479, −2.34718675546875141238127534133, −1.38160626974763524931763379696, 0,
1.38160626974763524931763379696, 2.34718675546875141238127534133, 3.24137434564808265370087117479, 4.09438452877911047401406976138, 4.70321741421989721958914979215, 5.54114747838435288152941825677, 6.04773962213833599570492747308, 6.60220618397810332038818813541, 7.47415218255942471976438176842