| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 3·11-s − 12-s − 13-s − 15-s + 16-s − 2·17-s − 18-s + 20-s + 3·22-s + 6·23-s + 24-s − 4·25-s + 26-s − 27-s + 9·29-s + 30-s + 5·31-s − 32-s + 3·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 1.67·29-s + 0.182·30-s + 0.898·31-s − 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−8.243607429146043504522013874167, −7.32832701445581839394281421194, −6.73829528137848242304816800129, −6.00176500482537992108642877854, −5.17829994705618640309208001418, −4.54468897104027052613567158098, −3.16759017860478469733364835165, −2.35496285556071794795478076294, −1.26138214262884201371726577271, 0,
1.26138214262884201371726577271, 2.35496285556071794795478076294, 3.16759017860478469733364835165, 4.54468897104027052613567158098, 5.17829994705618640309208001418, 6.00176500482537992108642877854, 6.73829528137848242304816800129, 7.32832701445581839394281421194, 8.243607429146043504522013874167
