| L(s) = 1 | − 2·2-s + 3·4-s − 4·7-s − 4·8-s − 2·9-s − 4·11-s + 8·14-s + 5·16-s + 4·18-s + 8·22-s − 16·23-s − 6·25-s − 12·28-s − 6·32-s − 6·36-s + 4·37-s − 24·43-s − 12·44-s + 32·46-s + 9·49-s + 12·50-s − 8·53-s + 16·56-s + 8·63-s + 7·64-s − 4·67-s + 16·71-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.51·7-s − 1.41·8-s − 2/3·9-s − 1.20·11-s + 2.13·14-s + 5/4·16-s + 0.942·18-s + 1.70·22-s − 3.33·23-s − 6/5·25-s − 2.26·28-s − 1.06·32-s − 36-s + 0.657·37-s − 3.65·43-s − 1.80·44-s + 4.71·46-s + 9/7·49-s + 1.69·50-s − 1.09·53-s + 2.13·56-s + 1.00·63-s + 7/8·64-s − 0.488·67-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.067375928988433787195037958255, −8.024357451025613311326857967234, −7.79562900233629194237685551305, −6.70748925180204914131686110030, −6.67990463469463645458853984117, −6.05374842428966131261230810044, −5.69856539643665466176583402943, −5.14813294034674159793348274299, −4.12037934811057976891599523144, −3.54075497823589975288034605375, −3.01547776594839914177937202099, −2.31842582984328713370689289895, −1.79345632998845375089907987040, 0, 0,
1.79345632998845375089907987040, 2.31842582984328713370689289895, 3.01547776594839914177937202099, 3.54075497823589975288034605375, 4.12037934811057976891599523144, 5.14813294034674159793348274299, 5.69856539643665466176583402943, 6.05374842428966131261230810044, 6.67990463469463645458853984117, 6.70748925180204914131686110030, 7.79562900233629194237685551305, 8.024357451025613311326857967234, 8.067375928988433787195037958255
