L(s) = 1 | − 2·2-s + 4-s + 3·5-s − 2·7-s + 2·8-s + 2·9-s − 6·10-s + 4·14-s − 3·16-s − 2·17-s − 4·18-s + 2·19-s + 3·20-s − 6·23-s + 2·25-s − 2·28-s − 2·29-s − 2·32-s + 4·34-s − 6·35-s + 2·36-s − 4·38-s + 6·40-s + 6·45-s + 12·46-s − 3·49-s − 4·50-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s + 0.707·8-s + 2/3·9-s − 1.89·10-s + 1.06·14-s − 3/4·16-s − 0.485·17-s − 0.942·18-s + 0.458·19-s + 0.670·20-s − 1.25·23-s + 2/5·25-s − 0.377·28-s − 0.371·29-s − 0.353·32-s + 0.685·34-s − 1.01·35-s + 1/3·36-s − 0.648·38-s + 0.948·40-s + 0.894·45-s + 1.76·46-s − 3/7·49-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259210 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.808608443538808251354117830777, −8.374660761514301807827180740251, −7.82177158082309574256546549172, −7.42165436018077563266466384530, −6.84126885728970177680822682370, −6.44676347761932085205425361424, −5.81008977969516010210793049103, −5.54086427102039865513295843511, −4.71298591391043228555796461260, −4.16491293181815754578700719815, −3.47947094828248928852420720767, −2.61919224324054466118126730153, −1.86927148122030931172765021999, −1.33823459909947567590692353753, 0,
1.33823459909947567590692353753, 1.86927148122030931172765021999, 2.61919224324054466118126730153, 3.47947094828248928852420720767, 4.16491293181815754578700719815, 4.71298591391043228555796461260, 5.54086427102039865513295843511, 5.81008977969516010210793049103, 6.44676347761932085205425361424, 6.84126885728970177680822682370, 7.42165436018077563266466384530, 7.82177158082309574256546549172, 8.374660761514301807827180740251, 8.808608443538808251354117830777