| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 3·13-s − 15-s + 17-s + 19-s + 4·21-s − 3·23-s − 4·25-s + 27-s + 10·29-s − 6·31-s − 4·35-s − 4·37-s − 3·39-s − 5·41-s − 43-s − 45-s + 2·47-s + 9·49-s + 51-s − 14·53-s + 57-s + 6·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.832·13-s − 0.258·15-s + 0.242·17-s + 0.229·19-s + 0.872·21-s − 0.625·23-s − 4/5·25-s + 0.192·27-s + 1.85·29-s − 1.07·31-s − 0.676·35-s − 0.657·37-s − 0.480·39-s − 0.780·41-s − 0.152·43-s − 0.149·45-s + 0.291·47-s + 9/7·49-s + 0.140·51-s − 1.92·53-s + 0.132·57-s + 0.781·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.17316890614247, −13.72526601720728, −13.04313455883691, −12.46312239719701, −11.94956507000308, −11.69251126649552, −11.15793942018181, −10.48873796942393, −10.11381992456730, −9.603018626419048, −8.833711382664570, −8.526370083479846, −7.976255224957769, −7.561251303225814, −7.266795597401592, −6.454145378190574, −5.836859955861499, −5.024566078850655, −4.803164800542946, −4.236138281837934, −3.530077350878361, −3.011946826027756, −2.115083850663034, −1.812820827146796, −1.002332455622238, 0,
1.002332455622238, 1.812820827146796, 2.115083850663034, 3.011946826027756, 3.530077350878361, 4.236138281837934, 4.803164800542946, 5.024566078850655, 5.836859955861499, 6.454145378190574, 7.266795597401592, 7.561251303225814, 7.976255224957769, 8.526370083479846, 8.833711382664570, 9.603018626419048, 10.11381992456730, 10.48873796942393, 11.15793942018181, 11.69251126649552, 11.94956507000308, 12.46312239719701, 13.04313455883691, 13.72526601720728, 14.17316890614247
