L(s) = 1 | − 5-s + 7-s − 3·9-s − 2·11-s + 2·13-s − 3·17-s − 6·19-s − 4·25-s − 7·29-s − 8·31-s − 35-s − 2·37-s − 6·41-s − 5·43-s + 3·45-s − 11·47-s − 6·49-s − 3·53-s + 2·55-s − 8·59-s − 2·61-s − 3·63-s − 2·65-s − 67-s + 3·71-s − 15·73-s − 2·77-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.603·11-s + 0.554·13-s − 0.727·17-s − 1.37·19-s − 4/5·25-s − 1.29·29-s − 1.43·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s − 0.762·43-s + 0.447·45-s − 1.60·47-s − 6/7·49-s − 0.412·53-s + 0.269·55-s − 1.04·59-s − 0.256·61-s − 0.377·63-s − 0.248·65-s − 0.122·67-s + 0.356·71-s − 1.75·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 1879 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.49264576198112, −15.14865471793058, −14.68053335466273, −14.16895314775355, −13.43002285729715, −13.07021144459145, −12.59419873252504, −11.71111512035572, −11.40130421632871, −10.94640010470868, −10.49686234190001, −9.707043656132006, −9.001314803793929, −8.572005528615778, −8.085796597583425, −7.580817867654262, −6.826384587824603, −6.172140619208368, −5.680360560061637, −4.959533014953050, −4.410624319995202, −3.558825558424635, −3.175550492600740, −2.052540754446849, −1.742063052905780, 0, 0,
1.742063052905780, 2.052540754446849, 3.175550492600740, 3.558825558424635, 4.410624319995202, 4.959533014953050, 5.680360560061637, 6.172140619208368, 6.826384587824603, 7.580817867654262, 8.085796597583425, 8.572005528615778, 9.001314803793929, 9.707043656132006, 10.49686234190001, 10.94640010470868, 11.40130421632871, 11.71111512035572, 12.59419873252504, 13.07021144459145, 13.43002285729715, 14.16895314775355, 14.68053335466273, 15.14865471793058, 15.49264576198112