| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s + 3·11-s − 5·13-s + 16-s − 2·17-s + 3·18-s + 5·19-s − 3·22-s + 7·23-s + 5·26-s − 4·29-s + 2·31-s − 32-s + 2·34-s − 3·36-s − 37-s − 5·38-s − 3·41-s − 2·43-s + 3·44-s − 7·46-s − 7·47-s − 5·52-s − 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 0.904·11-s − 1.38·13-s + 1/4·16-s − 0.485·17-s + 0.707·18-s + 1.14·19-s − 0.639·22-s + 1.45·23-s + 0.980·26-s − 0.742·29-s + 0.359·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s − 0.164·37-s − 0.811·38-s − 0.468·41-s − 0.304·43-s + 0.452·44-s − 1.03·46-s − 1.02·47-s − 0.693·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.722288200303847181464711533795, −7.80471099931965682898184894578, −7.12191173900728537243214164656, −6.43773390159168975843176858276, −5.43945201706455509668151309560, −4.71835791677761993355466214212, −3.36202621499620939854951938574, −2.65167937662424348651501651075, −1.42353746843223523996833758028, 0,
1.42353746843223523996833758028, 2.65167937662424348651501651075, 3.36202621499620939854951938574, 4.71835791677761993355466214212, 5.43945201706455509668151309560, 6.43773390159168975843176858276, 7.12191173900728537243214164656, 7.80471099931965682898184894578, 8.722288200303847181464711533795
