| L(s) = 1 | − 2.32i·5-s − i·7-s − 0.908i·11-s + (−1.32 − 3.35i)13-s − 1.93·17-s + 4.19i·19-s + 1.41·23-s − 0.419·25-s + 7.75·29-s − 3.87i·31-s − 2.32·35-s − 11.9i·37-s − 6.19i·41-s − 6.91·43-s + 3.16i·47-s + ⋯ |
| L(s) = 1 | − 1.04i·5-s − 0.377i·7-s − 0.273i·11-s + (−0.368 − 0.929i)13-s − 0.468·17-s + 0.961i·19-s + 0.295·23-s − 0.0838·25-s + 1.44·29-s − 0.695i·31-s − 0.393·35-s − 1.97i·37-s − 0.968i·41-s − 1.05·43-s + 0.462i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120537200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120537200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (1.32 + 3.35i)T \) |
| good | 5 | \( 1 + 2.32iT - 5T^{2} \) |
| 11 | \( 1 + 0.908iT - 11T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 4.19iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 7.75T + 29T^{2} \) |
| 31 | \( 1 + 3.87iT - 31T^{2} \) |
| 37 | \( 1 + 11.9iT - 37T^{2} \) |
| 41 | \( 1 + 6.19iT - 41T^{2} \) |
| 43 | \( 1 + 6.91T + 43T^{2} \) |
| 47 | \( 1 - 3.16iT - 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 4.97iT - 59T^{2} \) |
| 61 | \( 1 + 0.0483T + 61T^{2} \) |
| 67 | \( 1 + 3.76iT - 67T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 - 7.16iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 - 11.7iT - 89T^{2} \) |
| 97 | \( 1 - 6.37iT - 97T^{2} \) |
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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
−8.264029854260073140592375652281, −7.77368964572353663476173763135, −6.84762157409560247997729733270, −5.92111355338624973191352996807, −5.23550705277991893029180770264, −4.49490076306865935026856107007, −3.67063405331076510340095833844, −2.60973049736296790119946064418, −1.37637974374407496676731427861, −0.34562550069372353586130543541,
1.54774990506521917859998408974, 2.69180527314960563627560834455, 3.15595059815724933436593019100, 4.53520419526360059397432771017, 4.93755364249070895421740923103, 6.29549377461321243007543112424, 6.70642456043420696992974056442, 7.24140458977580950617048636838, 8.348651266755775118996114309817, 8.868890778665855734657795536641
