| L(s) = 1 | + (1.55 − 1.60i)5-s − 1.21·7-s − 2.73·11-s + 6.99i·13-s − 0.391i·17-s − 2.94i·19-s + (−3.74 − 3.00i)23-s + (−0.138 − 4.99i)25-s + 3.30i·29-s + 7.96·31-s + (−1.89 + 1.94i)35-s − 1.84·37-s − 9.35i·41-s + 4.89·43-s − 2.87·47-s + ⋯ |
| L(s) = 1 | + (0.697 − 0.716i)5-s − 0.459·7-s − 0.825·11-s + 1.94i·13-s − 0.0948i·17-s − 0.675i·19-s + (−0.779 − 0.625i)23-s + (−0.0277 − 0.999i)25-s + 0.613i·29-s + 1.42·31-s + (−0.320 + 0.329i)35-s − 0.303·37-s − 1.46i·41-s + 0.747·43-s − 0.419·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.116769420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.116769420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.55 + 1.60i)T \) |
| 23 | \( 1 + (3.74 + 3.00i)T \) |
| good | 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 6.99iT - 13T^{2} \) |
| 17 | \( 1 + 0.391iT - 17T^{2} \) |
| 19 | \( 1 + 2.94iT - 19T^{2} \) |
| 29 | \( 1 - 3.30iT - 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 + 1.84T + 37T^{2} \) |
| 41 | \( 1 + 9.35iT - 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 + 2.95iT - 53T^{2} \) |
| 59 | \( 1 + 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 1.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 + 12.9iT - 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
| 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
−8.302280003136181469125000774126, −7.43904696073271307632708777186, −6.45401161725254627419783695747, −6.21826388075267030292096785911, −4.92519587120032982725047647698, −4.69738421265758430411209676604, −3.57453073257339344924693185330, −2.41235593621769025701473911989, −1.76152339541428230916676316260, −0.31372054836268346409546387553,
1.21280899862427577901929492116, 2.63510144691607164001934711522, 2.94014179077167943504120833474, 3.97592752588188318002697584933, 5.15986902249326561482544475657, 5.84594423393275242238398137791, 6.22124811782320791073809983771, 7.29949927505893608155488079054, 7.906477501554267411323145725738, 8.463811858012427012425489755291
