| L(s) = 1 | + (−1.14 + 1.30i)3-s + (1.41 − 2.23i)7-s + (−0.381 − 2.97i)9-s + 1.13i·11-s + 0.333·13-s + 4.20i·17-s − 1.95i·19-s + (1.28 + 4.39i)21-s − 2.54·23-s + (4.30 + 2.90i)27-s − 8.49i·29-s − 5.11i·31-s + (−1.47 − 1.30i)33-s − 2.23i·37-s + (−0.381 + 0.434i)39-s + ⋯ |
| L(s) = 1 | + (−0.660 + 0.750i)3-s + (0.534 − 0.845i)7-s + (−0.127 − 0.991i)9-s + 0.342i·11-s + 0.0925·13-s + 1.02i·17-s − 0.448i·19-s + (0.281 + 0.959i)21-s − 0.529·23-s + (0.828 + 0.559i)27-s − 1.57i·29-s − 0.918i·31-s + (−0.257 − 0.226i)33-s − 0.367i·37-s + (−0.0611 + 0.0695i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175648102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175648102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.14 - 1.30i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
| good | 11 | \( 1 - 1.13iT - 11T^{2} \) |
| 13 | \( 1 - 0.333T + 13T^{2} \) |
| 17 | \( 1 - 4.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.95iT - 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 + 8.49iT - 29T^{2} \) |
| 31 | \( 1 + 5.11iT - 31T^{2} \) |
| 37 | \( 1 + 2.23iT - 37T^{2} \) |
| 41 | \( 1 - 5.81T + 41T^{2} \) |
| 43 | \( 1 - 0.527iT - 43T^{2} \) |
| 47 | \( 1 - 7.42iT - 47T^{2} \) |
| 53 | \( 1 + 8.22T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 6.70iT - 67T^{2} \) |
| 71 | \( 1 + 10.7iT - 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 1.47T + 79T^{2} \) |
| 83 | \( 1 + 5.81iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
| 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
−9.259504648907103336069473440333, −8.101524122753963582394711821987, −7.57231496758089842108885428392, −6.39757705353632154332703852718, −5.93440855043668068558969753933, −4.73532152840503296750152956656, −4.29770554344156519632380723574, −3.43971418747134506908018131544, −1.93660646134244854150906319141, −0.51292630124451714316640098298,
1.14115140487251036010074380555, 2.18468683051140167739567715796, 3.19899055519570272750824179241, 4.64789554520859864769917195017, 5.34589669385185705885191110987, 5.97665599276894257985823008536, 6.87631813060025602845444017024, 7.57641752450883395051707269514, 8.425245498487613553751571575412, 8.983046042737997229465468217619
