L(s) = 1 | − 5-s + 3·7-s + 2·11-s − 7·13-s + 2·17-s + 19-s − 6·23-s + 25-s + 9·29-s − 31-s − 3·35-s − 2·37-s + 3·41-s − 43-s − 8·47-s + 2·49-s + 10·53-s − 2·55-s + 4·59-s + 5·61-s + 7·65-s − 3·67-s − 12·71-s − 3·73-s + 6·77-s + 5·79-s − 16·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.13·7-s + 0.603·11-s − 1.94·13-s + 0.485·17-s + 0.229·19-s − 1.25·23-s + 1/5·25-s + 1.67·29-s − 0.179·31-s − 0.507·35-s − 0.328·37-s + 0.468·41-s − 0.152·43-s − 1.16·47-s + 2/7·49-s + 1.37·53-s − 0.269·55-s + 0.520·59-s + 0.640·61-s + 0.868·65-s − 0.366·67-s − 1.42·71-s − 0.351·73-s + 0.683·77-s + 0.562·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.09346251984029, −13.30190232624332, −12.66114321537153, −12.13051131488698, −11.81440291717715, −11.66078810940224, −10.91140699900194, −10.25597841131270, −9.973006513755010, −9.515683621285648, −8.681401626870089, −8.408873856425494, −7.810607998729457, −7.418277656214747, −6.976582225777024, −6.339590680213490, −5.626207142013760, −5.122117436778863, −4.576217719169470, −4.290188264545393, −3.521624628842920, −2.815713835959661, −2.236571965351311, −1.610178001868562, −0.8660569373524143, 0,
0.8660569373524143, 1.610178001868562, 2.236571965351311, 2.815713835959661, 3.521624628842920, 4.290188264545393, 4.576217719169470, 5.122117436778863, 5.626207142013760, 6.339590680213490, 6.976582225777024, 7.418277656214747, 7.810607998729457, 8.408873856425494, 8.681401626870089, 9.515683621285648, 9.973006513755010, 10.25597841131270, 10.91140699900194, 11.66078810940224, 11.81440291717715, 12.13051131488698, 12.66114321537153, 13.30190232624332, 14.09346251984029