L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s + 12-s − 6·13-s + 14-s − 16-s + 2·17-s + 18-s + 8·19-s − 21-s − 8·23-s + 3·24-s − 6·26-s − 27-s − 28-s + 2·29-s + 4·31-s + 5·32-s + 2·34-s − 36-s + 2·37-s + 8·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.218·21-s − 1.66·23-s + 0.612·24-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s + 0.342·34-s − 1/6·36-s + 0.328·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.41176994506969, −14.10989360222402, −13.55409729812714, −12.92525057968236, −12.43320022047063, −12.00941989372019, −11.70262748697354, −11.24659858256940, −10.18469896148672, −9.984825009352070, −9.584005381837561, −8.967055866667385, −8.168582457912486, −7.646728369464692, −7.350409640866583, −6.433859503458820, −5.919236810693479, −5.431165490993412, −4.906504501802016, −4.511668319539569, −3.933265169347224, −3.091344533097303, −2.660088166790085, −1.688225576602508, −0.8132522041137054, 0,
0.8132522041137054, 1.688225576602508, 2.660088166790085, 3.091344533097303, 3.933265169347224, 4.511668319539569, 4.906504501802016, 5.431165490993412, 5.919236810693479, 6.433859503458820, 7.350409640866583, 7.646728369464692, 8.168582457912486, 8.967055866667385, 9.584005381837561, 9.984825009352070, 10.18469896148672, 11.24659858256940, 11.70262748697354, 12.00941989372019, 12.43320022047063, 12.92525057968236, 13.55409729812714, 14.10989360222402, 14.41176994506969