| L(s) = 1 | + 2·2-s + 4-s − 4·5-s − 8·10-s − 2·11-s − 4·13-s + 16-s − 6·17-s − 6·19-s − 4·20-s − 4·22-s + 14·23-s + 2·25-s − 8·26-s + 2·29-s − 2·32-s − 12·34-s − 2·37-s − 12·38-s − 8·41-s − 14·43-s − 2·44-s + 28·46-s + 14·47-s + 4·50-s − 4·52-s + 20·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 1.78·5-s − 2.52·10-s − 0.603·11-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 1.37·19-s − 0.894·20-s − 0.852·22-s + 2.91·23-s + 2/5·25-s − 1.56·26-s + 0.371·29-s − 0.353·32-s − 2.05·34-s − 0.328·37-s − 1.94·38-s − 1.24·41-s − 2.13·43-s − 0.301·44-s + 4.12·46-s + 2.04·47-s + 0.565·50-s − 0.554·52-s + 2.74·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23532201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 41 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 135 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 95 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 183 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.013310597149278640594497958503, −7.57085489830467025490857425681, −7.32874370818733526107216500160, −6.96315287568312212206374229403, −6.72338240268617345048386194246, −6.40352154193492582690094313220, −5.49208405621410475664084915989, −5.48365316375383930385240237846, −4.87376010719391158869986270644, −4.80617143908473956648464794350, −4.34628880272666651491357226532, −4.06552336825994382101818226662, −3.68996633900787733164742912888, −3.35874488424920387228680921399, −2.69160792032119484006769477906, −2.51037583998353761264740630262, −1.87506268527371296560263131990, −1.01277526789946639591207650809, 0, 0,
1.01277526789946639591207650809, 1.87506268527371296560263131990, 2.51037583998353761264740630262, 2.69160792032119484006769477906, 3.35874488424920387228680921399, 3.68996633900787733164742912888, 4.06552336825994382101818226662, 4.34628880272666651491357226532, 4.80617143908473956648464794350, 4.87376010719391158869986270644, 5.48365316375383930385240237846, 5.49208405621410475664084915989, 6.40352154193492582690094313220, 6.72338240268617345048386194246, 6.96315287568312212206374229403, 7.32874370818733526107216500160, 7.57085489830467025490857425681, 8.013310597149278640594497958503
