| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s + 8-s + 9-s + 4·10-s − 11-s − 12-s + 13-s − 4·15-s + 16-s − 3·17-s + 18-s + 19-s + 4·20-s − 22-s + 6·23-s − 24-s + 11·25-s + 26-s − 27-s − 9·29-s − 4·30-s + 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s − 1.03·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.894·20-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s − 1.67·29-s − 0.730·30-s + 1.43·31-s + 0.176·32-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
| \(L(1)\) |
\(\approx\) |
\(3.689321626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.689321626\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.717563520607241339464403872223, −7.37824254385187506478383866516, −6.80466059046589804863171278808, −6.04861681701205841990889272290, −5.52890808221543797285284099235, −4.99437204761653671419451653807, −4.03221063148946237330637456826, −2.81769765180132857870073169744, −2.10938404257895940389241977355, −1.09932921028451441773943649645,
1.09932921028451441773943649645, 2.10938404257895940389241977355, 2.81769765180132857870073169744, 4.03221063148946237330637456826, 4.99437204761653671419451653807, 5.52890808221543797285284099235, 6.04861681701205841990889272290, 6.80466059046589804863171278808, 7.37824254385187506478383866516, 8.717563520607241339464403872223
