| L(s) = 1 | − 2-s − 4-s − 4·5-s − 7-s + 3·8-s + 9-s + 4·10-s + 8·11-s − 4·13-s + 14-s − 16-s − 18-s + 4·20-s − 8·22-s + 2·25-s + 4·26-s + 28-s − 5·32-s + 4·35-s − 36-s − 12·40-s − 8·43-s − 8·44-s − 4·45-s + 49-s − 2·50-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 1.26·10-s + 2.41·11-s − 1.10·13-s + 0.267·14-s − 1/4·16-s − 0.235·18-s + 0.894·20-s − 1.70·22-s + 2/5·25-s + 0.784·26-s + 0.188·28-s − 0.883·32-s + 0.676·35-s − 1/6·36-s − 1.89·40-s − 1.21·43-s − 1.20·44-s − 0.596·45-s + 1/7·49-s − 0.282·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197568 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.672365196683144779961098572501, −8.628787276523753003004430017583, −7.86869718610037511648213775056, −7.44376428871628094569671391724, −7.25047783802838427330028450541, −6.55597781799330496824615300000, −6.19439119239525436838633052918, −5.16773217950502102961008704567, −4.64274308636669756274291953837, −4.13559084050773741974089362056, −3.75545050483136750219295120519, −3.35852253915733761610986992002, −2.03425588439014896949448393282, −1.08543961772187534196881977330, 0,
1.08543961772187534196881977330, 2.03425588439014896949448393282, 3.35852253915733761610986992002, 3.75545050483136750219295120519, 4.13559084050773741974089362056, 4.64274308636669756274291953837, 5.16773217950502102961008704567, 6.19439119239525436838633052918, 6.55597781799330496824615300000, 7.25047783802838427330028450541, 7.44376428871628094569671391724, 7.86869718610037511648213775056, 8.628787276523753003004430017583, 8.672365196683144779961098572501
