| L(s) = 1 | − 54·5-s − 49·7-s + 594·11-s + 26·13-s + 534·17-s + 3.00e3·19-s + 3.51e3·23-s − 209·25-s − 4.29e3·29-s − 8.03e3·31-s + 2.64e3·35-s − 502·37-s − 9.87e3·41-s − 9.06e3·43-s + 1.14e3·47-s + 2.40e3·49-s − 2.83e4·53-s − 3.20e4·55-s − 8.19e3·59-s + 2.98e4·61-s − 1.40e3·65-s + 6.28e4·67-s − 3.43e4·71-s + 5.69e4·73-s − 2.91e4·77-s − 4.94e4·79-s − 5.25e4·83-s + ⋯ |
| L(s) = 1 | − 0.965·5-s − 0.377·7-s + 1.48·11-s + 0.0426·13-s + 0.448·17-s + 1.90·19-s + 1.38·23-s − 0.0668·25-s − 0.948·29-s − 1.50·31-s + 0.365·35-s − 0.0602·37-s − 0.916·41-s − 0.747·43-s + 0.0752·47-s + 1/7·49-s − 1.38·53-s − 1.42·55-s − 0.306·59-s + 1.02·61-s − 0.0412·65-s + 1.71·67-s − 0.809·71-s + 1.25·73-s − 0.559·77-s − 0.892·79-s − 0.836·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.886324810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.886324810\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 54 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 534 T + p^{5} T^{2} \) |
| 19 | \( 1 - 3004 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3510 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4296 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8036 T + p^{5} T^{2} \) |
| 37 | \( 1 + 502 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9870 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9068 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1140 T + p^{5} T^{2} \) |
| 53 | \( 1 + 28356 T + p^{5} T^{2} \) |
| 59 | \( 1 + 8196 T + p^{5} T^{2} \) |
| 61 | \( 1 - 29822 T + p^{5} T^{2} \) |
| 67 | \( 1 - 62884 T + p^{5} T^{2} \) |
| 71 | \( 1 + 34398 T + p^{5} T^{2} \) |
| 73 | \( 1 - 56990 T + p^{5} T^{2} \) |
| 79 | \( 1 + 49496 T + p^{5} T^{2} \) |
| 83 | \( 1 + 52512 T + p^{5} T^{2} \) |
| 89 | \( 1 - 48282 T + p^{5} T^{2} \) |
| 97 | \( 1 + 83938 T + p^{5} T^{2} \) |
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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
−9.316392675405860967905180987172, −8.429357749145049069092084673808, −7.36402692958941036896942375320, −6.98298830353660439146196105518, −5.79084316369725150780973269640, −4.86063003600800227114046603581, −3.59201993988522412643941056474, −3.37398415688373120613662412353, −1.60773222822976819283287064701, −0.62947526342165278129347517870,
0.62947526342165278129347517870, 1.60773222822976819283287064701, 3.37398415688373120613662412353, 3.59201993988522412643941056474, 4.86063003600800227114046603581, 5.79084316369725150780973269640, 6.98298830353660439146196105518, 7.36402692958941036896942375320, 8.429357749145049069092084673808, 9.316392675405860967905180987172
