| L(s) = 1 | + (−0.311 − 2.21i)5-s − i·7-s + 2·11-s − 6.42i·13-s + 4.42i·17-s − 2.42·19-s + 1.37i·23-s + (−4.80 + 1.37i)25-s + 0.755·29-s − 5.18·31-s + (−2.21 + 0.311i)35-s − 7.61i·37-s + 8.23·41-s − 10.1i·43-s − 2.75i·47-s + ⋯ |
| L(s) = 1 | + (−0.139 − 0.990i)5-s − 0.377i·7-s + 0.603·11-s − 1.78i·13-s + 1.07i·17-s − 0.557·19-s + 0.287i·23-s + (−0.961 + 0.275i)25-s + 0.140·29-s − 0.931·31-s + (−0.374 + 0.0525i)35-s − 1.25i·37-s + 1.28·41-s − 1.54i·43-s − 0.401i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.045285329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.045285329\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.311 + 2.21i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6.42iT - 13T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 23 | \( 1 - 1.37iT - 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 + 5.18T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 2.75iT - 47T^{2} \) |
| 53 | \( 1 + 9.18iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 - 2.75iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 1.57iT - 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 4.62T + 89T^{2} \) |
| 97 | \( 1 - 11.9iT - 97T^{2} \) |
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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
−7.921112032623520072394939779410, −7.38407017215259363469654163276, −6.33273442881667401778858033944, −5.61631688695792549556463640737, −5.07819874837035652056315413293, −3.93895573677988932253739247407, −3.67513301535098410227844086987, −2.29593133679340886196695142038, −1.25428320123999939157933123893, −0.28646285925087774487799971726,
1.49768388564283524996941479045, 2.44211054645320300142899831148, 3.17736154454198428490515862420, 4.22269158455393790350914564583, 4.68367863139862182232691761632, 5.97787865124032797632357732830, 6.43661333362691415893509575889, 7.08350055902717716486260342925, 7.69579506575935538218609106189, 8.679725018680887132771159219251
