L(s) = 1 | + (1.53 + 0.5i)2-s + (1.30 + 0.951i)4-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (−1.30 + 0.951i)14-s + (−0.951 + 1.30i)18-s + 1.61i·22-s − 1.61i·23-s + (−1.53 + 0.5i)28-s + (0.5 + 0.363i)29-s − 0.999i·32-s + (−1.30 + 0.951i)36-s + (0.951 − 1.30i)37-s + ⋯ |
L(s) = 1 | + (1.53 + 0.5i)2-s + (1.30 + 0.951i)4-s + (−0.587 + 0.809i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (−1.30 + 0.951i)14-s + (−0.951 + 1.30i)18-s + 1.61i·22-s − 1.61i·23-s + (−1.53 + 0.5i)28-s + (0.5 + 0.363i)29-s − 0.999i·32-s + (−1.30 + 0.951i)36-s + (0.951 − 1.30i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.448891946\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.448891946\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + 0.618iT - T^{2} \) |
| 71 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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
−9.534019385049389086397655093252, −8.663716549383187175629886905454, −7.74108125973498438906952137116, −6.89608001204811573067431470532, −6.25377313361782827128508369108, −5.48522239952156107844755215786, −4.75464831125678761928993075674, −4.06045586634432769015305571174, −2.84175185020310661442396360288, −2.25262430652730879199032266276,
1.18615282927778834576647626256, 2.77927374351845926050113090874, 3.52072321391452343036978048536, 3.97856431985012422388014972643, 5.06333178000084571900544388768, 6.01720828486222088777802734091, 6.39752573788718736289757695310, 7.34229009951508642500653837705, 8.467533470587452572409017052187, 9.386966481604063547857052383998