L(s) = 1 | + 4·2-s − 14·3-s + 16·4-s − 56·6-s − 158·7-s + 64·8-s − 47·9-s − 148·11-s − 224·12-s − 684·13-s − 632·14-s + 256·16-s − 2.04e3·17-s − 188·18-s + 2.22e3·19-s + 2.21e3·21-s − 592·22-s + 1.24e3·23-s − 896·24-s − 2.73e3·26-s + 4.06e3·27-s − 2.52e3·28-s − 270·29-s − 2.04e3·31-s + 1.02e3·32-s + 2.07e3·33-s − 8.19e3·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.898·3-s + 1/2·4-s − 0.635·6-s − 1.21·7-s + 0.353·8-s − 0.193·9-s − 0.368·11-s − 0.449·12-s − 1.12·13-s − 0.861·14-s + 1/4·16-s − 1.71·17-s − 0.136·18-s + 1.41·19-s + 1.09·21-s − 0.260·22-s + 0.491·23-s − 0.317·24-s − 0.793·26-s + 1.07·27-s − 0.609·28-s − 0.0596·29-s − 0.382·31-s + 0.176·32-s + 0.331·33-s − 1.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 14 T + p^{5} T^{2} \) |
| 7 | \( 1 + 158 T + p^{5} T^{2} \) |
| 11 | \( 1 + 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 684 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2220 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1246 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4372 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2398 T + p^{5} T^{2} \) |
| 43 | \( 1 + 2294 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10682 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2964 T + p^{5} T^{2} \) |
| 59 | \( 1 + 39740 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42298 T + p^{5} T^{2} \) |
| 67 | \( 1 + 32098 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 + 30104 T + p^{5} T^{2} \) |
| 79 | \( 1 - 35280 T + p^{5} T^{2} \) |
| 83 | \( 1 - 27826 T + p^{5} T^{2} \) |
| 89 | \( 1 + 85210 T + p^{5} T^{2} \) |
| 97 | \( 1 - 97232 T + p^{5} T^{2} \) |
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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
−13.76687182289615273782577049366, −12.76116946830580822704485861269, −11.80402499298806288614731212739, −10.69679989356991589984226517485, −9.347621679306487193878717081243, −7.19737613179483278704107044851, −6.08295761060109529882533052878, −4.83679840622951713964954782483, −2.88836097607258925743486871536, 0,
2.88836097607258925743486871536, 4.83679840622951713964954782483, 6.08295761060109529882533052878, 7.19737613179483278704107044851, 9.347621679306487193878717081243, 10.69679989356991589984226517485, 11.80402499298806288614731212739, 12.76116946830580822704485861269, 13.76687182289615273782577049366