| L(s) = 1 | − 2-s − 7-s + 8-s − 11-s − 2·13-s + 14-s − 16-s − 6·17-s − 2·19-s + 22-s − 6·23-s + 5·25-s + 2·26-s − 18·29-s + 4·31-s + 6·34-s − 2·37-s + 2·38-s + 12·41-s − 8·43-s + 6·46-s − 6·47-s − 6·49-s − 5·50-s − 56-s + 18·58-s − 3·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.377·7-s + 0.353·8-s − 0.301·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.458·19-s + 0.213·22-s − 1.25·23-s + 25-s + 0.392·26-s − 3.34·29-s + 0.718·31-s + 1.02·34-s − 0.328·37-s + 0.324·38-s + 1.87·41-s − 1.21·43-s + 0.884·46-s − 0.875·47-s − 6/7·49-s − 0.707·50-s − 0.133·56-s + 2.36·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.376460646078963123126576875486, −9.102094583012696668311044380984, −8.548265510824308634608129034699, −8.275245327040877334989810364762, −7.77603050703395187515679246367, −7.42453009145221053853016903332, −7.02169740466751354430693005481, −6.62319304182279089679844720155, −5.98173726066730259603915163837, −5.88450694678155946372616511428, −5.09235323048924129838600573715, −4.74616764085638489564831537355, −4.16539820694835223902198448717, −3.87903377406070274901156137094, −3.09290232622945152361582810853, −2.60234712402072617122097609065, −1.93629854569660215582082347188, −1.51054318668163971163136608735, 0, 0,
1.51054318668163971163136608735, 1.93629854569660215582082347188, 2.60234712402072617122097609065, 3.09290232622945152361582810853, 3.87903377406070274901156137094, 4.16539820694835223902198448717, 4.74616764085638489564831537355, 5.09235323048924129838600573715, 5.88450694678155946372616511428, 5.98173726066730259603915163837, 6.62319304182279089679844720155, 7.02169740466751354430693005481, 7.42453009145221053853016903332, 7.77603050703395187515679246367, 8.275245327040877334989810364762, 8.548265510824308634608129034699, 9.102094583012696668311044380984, 9.376460646078963123126576875486
