| L(s) = 1 | + 2·5-s − 2·11-s − 4·13-s − 4·17-s + 3·25-s − 4·29-s − 4·31-s − 4·37-s − 4·41-s + 8·43-s − 12·49-s − 20·53-s − 4·55-s − 4·59-s − 12·61-s − 8·65-s − 12·71-s + 12·73-s − 8·79-s − 8·83-s − 8·85-s − 20·89-s − 4·97-s + 4·101-s + 24·103-s − 8·107-s − 4·109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s − 1.10·13-s − 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.718·31-s − 0.657·37-s − 0.624·41-s + 1.21·43-s − 1.71·49-s − 2.74·53-s − 0.539·55-s − 0.520·59-s − 1.53·61-s − 0.992·65-s − 1.42·71-s + 1.40·73-s − 0.900·79-s − 0.878·83-s − 0.867·85-s − 2.11·89-s − 0.406·97-s + 0.398·101-s + 2.36·103-s − 0.773·107-s − 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 180 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 164 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.265513164557404834395079030213, −7.908970079033027260109667651765, −7.37982476357739626880923717120, −7.36232082944698600963901364124, −6.70625806993815846572572581033, −6.51074593252933263327192280547, −6.00566937903738438986698599382, −5.70372082677681664040108252280, −5.28057167260991902992444616535, −4.89672981378701668148221215481, −4.52255745184981330663156890322, −4.30217181647171831732431033724, −3.36950202238629285004491334713, −3.26759731766603956092872859718, −2.57361798734579017977230528527, −2.32677666498597041577469461569, −1.61609576815564888925979769689, −1.46443394382296823242613516156, 0, 0,
1.46443394382296823242613516156, 1.61609576815564888925979769689, 2.32677666498597041577469461569, 2.57361798734579017977230528527, 3.26759731766603956092872859718, 3.36950202238629285004491334713, 4.30217181647171831732431033724, 4.52255745184981330663156890322, 4.89672981378701668148221215481, 5.28057167260991902992444616535, 5.70372082677681664040108252280, 6.00566937903738438986698599382, 6.51074593252933263327192280547, 6.70625806993815846572572581033, 7.36232082944698600963901364124, 7.37982476357739626880923717120, 7.908970079033027260109667651765, 8.265513164557404834395079030213
