| L(s) = 1 | − 4·5-s + 2·7-s + 13-s − 7·17-s − 19-s + 6·23-s + 11·25-s + 4·29-s + 6·31-s − 8·35-s − 8·37-s + 6·41-s + 10·43-s + 4·47-s − 3·49-s + 13·53-s − 5·59-s − 2·61-s − 4·65-s − 14·67-s − 3·71-s + 10·73-s + 13·79-s + 15·83-s + 28·85-s + 7·89-s + 2·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.755·7-s + 0.277·13-s − 1.69·17-s − 0.229·19-s + 1.25·23-s + 11/5·25-s + 0.742·29-s + 1.07·31-s − 1.35·35-s − 1.31·37-s + 0.937·41-s + 1.52·43-s + 0.583·47-s − 3/7·49-s + 1.78·53-s − 0.650·59-s − 0.256·61-s − 0.496·65-s − 1.71·67-s − 0.356·71-s + 1.17·73-s + 1.46·79-s + 1.64·83-s + 3.03·85-s + 0.741·89-s + 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.58785971697598, −12.26677247708081, −11.99227719473187, −11.30407010736828, −11.14408163584451, −10.61967709547535, −10.50188820560428, −9.435936455234038, −8.970602002550853, −8.624007991452187, −8.327061389740686, −7.667034507665791, −7.420616830030803, −6.849056232732358, −6.477923120406955, −5.838746259124203, −4.882678475749656, −4.843391442497631, −4.255414969332720, −3.878651745786111, −3.270407826136502, −2.625197187357810, −2.189697761689505, −1.180662984270012, −0.7605071758556094, 0,
0.7605071758556094, 1.180662984270012, 2.189697761689505, 2.625197187357810, 3.270407826136502, 3.878651745786111, 4.255414969332720, 4.843391442497631, 4.882678475749656, 5.838746259124203, 6.477923120406955, 6.849056232732358, 7.420616830030803, 7.667034507665791, 8.327061389740686, 8.624007991452187, 8.970602002550853, 9.435936455234038, 10.50188820560428, 10.61967709547535, 11.14408163584451, 11.30407010736828, 11.99227719473187, 12.26677247708081, 12.58785971697598
