| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 14-s − 15-s + 16-s − 17-s + 18-s − 8·19-s − 20-s − 21-s + 22-s − 7·23-s + 24-s − 4·25-s + 27-s − 28-s − 7·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s − 1.45·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s − 0.188·28-s − 1.29·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.50210583615799, −13.14144813682819, −12.65714994642131, −12.36110770336784, −11.80877103928289, −11.17808082712638, −10.98008554821149, −10.27552947406773, −9.806541640416631, −9.386071639393786, −8.744844100777751, −8.187631147114751, −7.992964230932642, −7.242334795763366, −6.811285903378652, −6.314969462438851, −5.890014427605881, −5.204283754614694, −4.613690007037370, −4.060405206097186, −3.590208353429960, −3.412893837719415, −2.427192676493090, −1.877227887313606, −1.632237291918871, 0, 0,
1.632237291918871, 1.877227887313606, 2.427192676493090, 3.412893837719415, 3.590208353429960, 4.060405206097186, 4.613690007037370, 5.204283754614694, 5.890014427605881, 6.314969462438851, 6.811285903378652, 7.242334795763366, 7.992964230932642, 8.187631147114751, 8.744844100777751, 9.386071639393786, 9.806541640416631, 10.27552947406773, 10.98008554821149, 11.17808082712638, 11.80877103928289, 12.36110770336784, 12.65714994642131, 13.14144813682819, 13.50210583615799
