L(s) = 1 | + (−0.0646 + 0.198i)5-s + (1.47 − 1.07i)7-s + (0.309 + 0.951i)9-s + (−0.913 + 0.406i)11-s + (−0.604 + 1.86i)17-s + (0.809 + 0.587i)19-s − 0.618·23-s + (0.773 + 0.562i)25-s + (0.118 + 0.363i)35-s − 1.33·43-s − 0.209·45-s + (1.08 + 0.786i)47-s + (0.722 − 2.22i)49-s + (−0.0218 − 0.207i)55-s + (0.413 − 1.27i)61-s + ⋯ |
L(s) = 1 | + (−0.0646 + 0.198i)5-s + (1.47 − 1.07i)7-s + (0.309 + 0.951i)9-s + (−0.913 + 0.406i)11-s + (−0.604 + 1.86i)17-s + (0.809 + 0.587i)19-s − 0.618·23-s + (0.773 + 0.562i)25-s + (0.118 + 0.363i)35-s − 1.33·43-s − 0.209·45-s + (1.08 + 0.786i)47-s + (0.722 − 2.22i)49-s + (−0.0218 − 0.207i)55-s + (0.413 − 1.27i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.412204391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.412204391\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-1.47 + 1.07i)T + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.604 - 1.86i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + 0.618T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.33T + T^{2} \) |
| 47 | \( 1 + (-1.08 - 0.786i)T + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 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
−8.554535234583504697233712620705, −7.990920837611746732055160228107, −7.61508588824473221097305938245, −6.86083119018638253012385434748, −5.73153541186229024582972784641, −4.92309293100906155805269360818, −4.39456384997138807985853641633, −3.50965478291085018301487789022, −2.08974183355695739063359943758, −1.49079358451271958918693356124,
0.914355804684205294424322723752, 2.28714738806929781821899086723, 2.93060497224455802261274714088, 4.22500093902858812970833306934, 5.17320729446569768862363954395, 5.31716492993744211749463033568, 6.52724586426050709582865293083, 7.30973313839621953694747299463, 8.069634747853557740493888601220, 8.767519036661018329128693023669