| L(s) = 1 | − 3-s − 2·4-s + 9-s − 3·11-s + 2·12-s + 9·23-s − 9·25-s − 27-s − 3·31-s + 3·33-s − 2·36-s − 10·37-s + 6·44-s − 15·47-s − 3·49-s − 12·53-s − 3·59-s + 8·64-s − 12·67-s − 9·69-s + 12·71-s + 9·75-s + 81-s + 12·89-s − 18·92-s + 3·93-s − 5·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 1.87·23-s − 9/5·25-s − 0.192·27-s − 0.538·31-s + 0.522·33-s − 1/3·36-s − 1.64·37-s + 0.904·44-s − 2.18·47-s − 3/7·49-s − 1.64·53-s − 0.390·59-s + 64-s − 1.46·67-s − 1.08·69-s + 1.42·71-s + 1.03·75-s + 1/9·81-s + 1.27·89-s − 1.87·92-s + 0.311·93-s − 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29403 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29403 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−10.24236649283561907925013437606, −9.692664195911891194269346114894, −9.321841214306022199803039975701, −8.750219990406069608684507527653, −8.126898815412274783042438879530, −7.65688657973525918344057765146, −6.97471937239971577799516986717, −6.37995808510632478729456362038, −5.63074501778753162699395659731, −4.92292274486281330573182722756, −4.84534209112063407514340859686, −3.79222364194117200726223679417, −3.10090294262294957360333683300, −1.77967209060158706987625791996, 0,
1.77967209060158706987625791996, 3.10090294262294957360333683300, 3.79222364194117200726223679417, 4.84534209112063407514340859686, 4.92292274486281330573182722756, 5.63074501778753162699395659731, 6.37995808510632478729456362038, 6.97471937239971577799516986717, 7.65688657973525918344057765146, 8.126898815412274783042438879530, 8.750219990406069608684507527653, 9.321841214306022199803039975701, 9.692664195911891194269346114894, 10.24236649283561907925013437606
