| L(s) = 1 | + 3-s + 2·5-s + 9-s − 2·11-s + 2·15-s + 2·17-s − 5·19-s − 6·23-s − 25-s + 27-s − 2·29-s − 2·33-s + 5·37-s + 6·41-s + 7·43-s + 2·45-s + 2·47-s + 2·51-s + 2·53-s − 4·55-s − 5·57-s + 10·59-s − 5·61-s + 4·67-s − 6·69-s − 6·71-s + 7·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s + 0.485·17-s − 1.14·19-s − 1.25·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.348·33-s + 0.821·37-s + 0.937·41-s + 1.06·43-s + 0.298·45-s + 0.291·47-s + 0.280·51-s + 0.274·53-s − 0.539·55-s − 0.662·57-s + 1.30·59-s − 0.640·61-s + 0.488·67-s − 0.722·69-s − 0.712·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.261667012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261667012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−13.10450298199505, −12.74738034148573, −12.22306509138166, −11.77641716858087, −10.95151871128839, −10.74663106960990, −10.11492529935430, −9.794312891534366, −9.326041823667291, −8.895029732226499, −8.177090285486089, −7.970882399245877, −7.412591339000234, −6.827471757455130, −6.099980907102771, −5.935831584900959, −5.349853092972752, −4.701555036577950, −4.035768179005657, −3.767002109354569, −2.854301239855540, −2.323981650007770, −2.087243211649959, −1.293500857510660, −0.4770547209623212,
0.4770547209623212, 1.293500857510660, 2.087243211649959, 2.323981650007770, 2.854301239855540, 3.767002109354569, 4.035768179005657, 4.701555036577950, 5.349853092972752, 5.935831584900959, 6.099980907102771, 6.827471757455130, 7.412591339000234, 7.970882399245877, 8.177090285486089, 8.895029732226499, 9.326041823667291, 9.794312891534366, 10.11492529935430, 10.74663106960990, 10.95151871128839, 11.77641716858087, 12.22306509138166, 12.74738034148573, 13.10450298199505
