L(s) = 1 | + 1.13i·2-s + 2.60i·3-s + 0.714·4-s − 2.95·6-s + 0.407i·7-s + 3.07i·8-s − 3.77·9-s + 2·11-s + 1.86i·12-s + 0.700i·13-s − 0.461·14-s − 2.05·16-s + 1.58i·17-s − 4.27i·18-s − 4.95·19-s + ⋯ |
L(s) = 1 | + 0.801i·2-s + 1.50i·3-s + 0.357·4-s − 1.20·6-s + 0.153i·7-s + 1.08i·8-s − 1.25·9-s + 0.603·11-s + 0.537i·12-s + 0.194i·13-s − 0.123·14-s − 0.514·16-s + 0.383i·17-s − 1.00i·18-s − 1.13·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64243i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.13iT - 2T^{2} \) |
| 3 | \( 1 - 2.60iT - 3T^{2} \) |
| 7 | \( 1 - 0.407iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.700iT - 13T^{2} \) |
| 17 | \( 1 - 1.58iT - 17T^{2} \) |
| 19 | \( 1 + 4.95T + 19T^{2} \) |
| 23 | \( 1 + 1.20iT - 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 8.20T + 31T^{2} \) |
| 37 | \( 1 + 5.13iT - 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + 1.27iT - 47T^{2} \) |
| 53 | \( 1 - 5.07iT - 53T^{2} \) |
| 59 | \( 1 - 6.49T + 59T^{2} \) |
| 61 | \( 1 + 9.42T + 61T^{2} \) |
| 67 | \( 1 + 3.08iT - 67T^{2} \) |
| 71 | \( 1 - 6.86T + 71T^{2} \) |
| 73 | \( 1 + 0.545iT - 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 0.974iT - 83T^{2} \) |
| 89 | \( 1 - 2.26T + 89T^{2} \) |
| 97 | \( 1 - 15.2iT - 97T^{2} \) |
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
−10.84037638679330295394930333298, −10.25107370186037092678714682052, −9.129230515804583069805276856201, −8.616851546427381528741881821583, −7.48470591330196811149396912184, −6.39448704778335809200505855075, −5.66084830456793372486066564374, −4.59991724578776282217546614481, −3.76192568949082443731332114637, −2.31701566371914172604145336869,
0.917299398637956091300305279931, 1.97260617274964958684803338868, 2.94865898953985717761374478965, 4.28505296981207434430536691210, 6.00973534282986761089781211787, 6.64163123868274172149984134799, 7.41416904222313670743157074816, 8.264194786713195173881651501060, 9.409974644842109675578341579826, 10.37413867671529188858999425931