| L(s) = 1 | + i·2-s − 4-s − 1.06·5-s + (0.884 + 2.49i)7-s − i·8-s − 1.06i·10-s − i·11-s + (−2.49 + 0.884i)14-s + 16-s + 3.91·17-s + 7.49i·19-s + 1.06·20-s + 22-s − 5.33i·23-s − 3.85·25-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.478·5-s + (0.334 + 0.942i)7-s − 0.353i·8-s − 0.338i·10-s − 0.301i·11-s + (−0.666 + 0.236i)14-s + 0.250·16-s + 0.950·17-s + 1.71i·19-s + 0.239·20-s + 0.213·22-s − 1.11i·23-s − 0.771·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012115018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012115018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.884 - 2.49i)T \) |
| 11 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.06T + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 7.49iT - 19T^{2} \) |
| 23 | \( 1 + 5.33iT - 23T^{2} \) |
| 29 | \( 1 - 8.01iT - 29T^{2} \) |
| 31 | \( 1 - 3.65iT - 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 0.230T + 43T^{2} \) |
| 47 | \( 1 + 0.370T + 47T^{2} \) |
| 53 | \( 1 + 7.56iT - 53T^{2} \) |
| 59 | \( 1 + 2.13T + 59T^{2} \) |
| 61 | \( 1 - 14.2iT - 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 3.15iT - 71T^{2} \) |
| 73 | \( 1 - 0.370iT - 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
| 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
−9.900755128330339047252679913021, −8.750681646546180773827771073261, −8.383760920162532913147051937068, −7.61362598379776062083586054714, −6.67822853836865594931003511768, −5.71517554574045581962260309450, −5.20929474987850721043754999883, −4.01524361781203817235725187035, −3.10776495201116686686971568355, −1.56256071293657096943907100737,
0.42641383156806202489021768208, 1.75314653620608062226788221596, 3.08339268355336790263202978940, 3.98798202755529801348075037899, 4.70641723935617823793519562758, 5.70554757453599763909110716465, 7.00700342183080894423327494577, 7.64284258529484634471979375562, 8.364530258483650138777188317111, 9.516916193016445405175050284711
