| L(s) = 1 | + 2·5-s + 7-s − 4·11-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 8·31-s + 2·35-s + 2·37-s + 2·41-s − 4·43-s − 8·47-s + 49-s − 6·53-s − 8·55-s − 6·61-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s + 16·79-s + 8·83-s + 12·85-s − 6·89-s − 16·95-s + 6·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.20·11-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 0.768·61-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s + 1.80·79-s + 0.878·83-s + 1.30·85-s − 0.635·89-s − 1.64·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470752405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470752405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.94427954008100, −13.33685481177663, −12.85099079105821, −12.75280596596686, −11.99909832713037, −11.32990742650406, −10.89147368309974, −10.35235596811974, −10.10042840671862, −9.371557844804988, −9.052032100070287, −8.254909063668460, −7.844418889519811, −7.499321549630342, −6.668365327764732, −6.126622420787198, −5.600904244019957, −5.251960179855980, −4.631153178042096, −3.898614617041553, −3.267452436863713, −2.577407135010646, −1.873216436962159, −1.596405254158065, −0.3624220808841593,
0.3624220808841593, 1.596405254158065, 1.873216436962159, 2.577407135010646, 3.267452436863713, 3.898614617041553, 4.631153178042096, 5.251960179855980, 5.600904244019957, 6.126622420787198, 6.668365327764732, 7.499321549630342, 7.844418889519811, 8.254909063668460, 9.052032100070287, 9.371557844804988, 10.10042840671862, 10.35235596811974, 10.89147368309974, 11.32990742650406, 11.99909832713037, 12.75280596596686, 12.85099079105821, 13.33685481177663, 13.94427954008100
