| L(s) = 1 | + 0.381i·2-s − 0.381·3-s + 1.85·4-s + 0.381i·5-s − 0.145i·6-s − i·7-s + 1.47i·8-s − 2.85·9-s − 0.145·10-s + 4.85i·11-s − 0.708·12-s + 0.381·14-s − 0.145i·15-s + 3.14·16-s − 7.47·17-s − 1.09i·18-s + ⋯ |
| L(s) = 1 | + 0.270i·2-s − 0.220·3-s + 0.927·4-s + 0.170i·5-s − 0.0595i·6-s − 0.377i·7-s + 0.520i·8-s − 0.951·9-s − 0.0461·10-s + 1.46i·11-s − 0.204·12-s + 0.102·14-s − 0.0376i·15-s + 0.786·16-s − 1.81·17-s − 0.256i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.246640092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246640092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.381iT - 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 - 0.381iT - 5T^{2} \) |
| 11 | \( 1 - 4.85iT - 11T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 4.85iT - 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 - 8.70iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 5.23iT - 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + 2.23iT - 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 + 2.23iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 0.708iT - 67T^{2} \) |
| 71 | \( 1 - 8.18iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 16.0iT - 89T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30737031216582589267580490487, −9.172295110731108200752259678219, −8.294226197271392202762337285362, −7.37808534349143156826382213354, −6.77100475003764675349577429712, −6.03036648826980734905579628927, −5.04652312951570257693748430913, −3.98590698758697232839942244852, −2.67165943506258580191332089698, −1.78886827469097648380728087232,
0.49526435249754016949954181262, 2.25768545863623087212992431466, 2.92640210224646174079169296074, 4.13919805769599941666922376112, 5.46646404113471179812586400776, 6.11776812750300879810476337712, 6.79473931119028543903826281578, 7.938599164640052163964412404072, 8.757583941572562374845542417263, 9.327571051016234460041317624738
