| L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s + 9-s + 11-s + 2·14-s + 5·16-s − 2·18-s − 2·22-s − 9·23-s − 4·25-s − 3·28-s + 6·29-s − 6·32-s + 3·36-s − 5·37-s + 7·43-s + 3·44-s + 18·46-s − 6·49-s + 8·50-s + 6·53-s + 4·56-s − 12·58-s − 63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s + 0.301·11-s + 0.534·14-s + 5/4·16-s − 0.471·18-s − 0.426·22-s − 1.87·23-s − 4/5·25-s − 0.566·28-s + 1.11·29-s − 1.06·32-s + 1/2·36-s − 0.821·37-s + 1.06·43-s + 0.452·44-s + 2.65·46-s − 6/7·49-s + 1.13·50-s + 0.824·53-s + 0.534·56-s − 1.57·58-s − 0.125·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.507897524075165785759917404928, −8.445534035290414055305010958501, −7.77469473704195364077621229491, −7.44672813365688738975842756286, −6.87681975385504124914718793711, −6.45592963182456326471421682466, −5.95263483747271245182502541366, −5.56638299187402291856591788143, −4.64180248246595770637838520702, −4.03083308612443923525672623490, −3.46733874547757557355626504081, −2.64322027621761890600480234130, −2.02109411661187122780364708336, −1.23566699110471542844307728143, 0,
1.23566699110471542844307728143, 2.02109411661187122780364708336, 2.64322027621761890600480234130, 3.46733874547757557355626504081, 4.03083308612443923525672623490, 4.64180248246595770637838520702, 5.56638299187402291856591788143, 5.95263483747271245182502541366, 6.45592963182456326471421682466, 6.87681975385504124914718793711, 7.44672813365688738975842756286, 7.77469473704195364077621229491, 8.445534035290414055305010958501, 8.507897524075165785759917404928
