| L(s) = 1 | − i·2-s + i·3-s + 6-s − i·8-s + i·13-s − 16-s − i·23-s + 24-s + 26-s + i·27-s + 29-s − 31-s − 39-s − 41-s − 46-s − i·47-s + ⋯ |
| L(s) = 1 | − i·2-s + i·3-s + 6-s − i·8-s + i·13-s − 16-s − i·23-s + 24-s + 26-s + i·27-s + 29-s − 31-s − 39-s − 41-s − 46-s − i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.021294606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.021294606\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 2 | \( 1 + iT - T^{2} \) |
| 3 | \( 1 - iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
| 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
−10.77067463766286194891790003929, −10.18058992315420349617663149641, −9.456585724295044641256665549481, −8.652200409677998182953046443468, −7.20606564791886697847592394754, −6.36250787984513985217524162647, −4.86152338754019419567466716310, −4.08694676575196260781789254721, −3.11637946849736173689483765439, −1.77729498819638364327312025721,
1.68386444314566656807282380129, 3.08725306834997735310018264887, 4.82604822745653848080637946004, 5.85888635409380811003523097697, 6.56296413536121853986544185145, 7.51645198797023996243272444032, 7.86839768481314197275041051610, 8.895171688336434243859849142807, 10.11668434567482410329629745273, 11.08980278915461948893475849611
