| L(s) = 1 | − 2-s − 6·3-s + 4-s − 3·5-s + 6·6-s + 8-s + 21·9-s + 3·10-s − 6·12-s + 18·15-s − 3·16-s − 21·18-s − 3·20-s − 6·24-s + 8·25-s − 54·27-s − 10·29-s − 18·30-s + 5·32-s + 21·36-s − 3·40-s − 9·41-s − 63·45-s + 2·47-s + 18·48-s + 49-s − 8·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3.46·3-s + 1/2·4-s − 1.34·5-s + 2.44·6-s + 0.353·8-s + 7·9-s + 0.948·10-s − 1.73·12-s + 4.64·15-s − 3/4·16-s − 4.94·18-s − 0.670·20-s − 1.22·24-s + 8/5·25-s − 10.3·27-s − 1.85·29-s − 3.28·30-s + 0.883·32-s + 7/2·36-s − 0.474·40-s − 1.40·41-s − 9.39·45-s + 0.291·47-s + 2.59·48-s + 1/7·49-s − 1.13·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 580810 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580810 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 241 | $C_2$ | \( 1 - 14 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 147 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.83413391287792649953413884502, −7.34279883968137600047389740645, −6.91673632582524755794371406427, −6.71097907961502948186435297164, −6.18394139476048934919468596978, −5.69474235311025643906987264660, −5.24071644967654257014345063377, −4.79093879727386134054263748555, −4.39412597791935874978981170167, −3.98270507636089794903823653142, −3.11227826530170626910414334567, −1.66870010080441215027671990268, −1.21552367676988469313364654996, 0, 0,
1.21552367676988469313364654996, 1.66870010080441215027671990268, 3.11227826530170626910414334567, 3.98270507636089794903823653142, 4.39412597791935874978981170167, 4.79093879727386134054263748555, 5.24071644967654257014345063377, 5.69474235311025643906987264660, 6.18394139476048934919468596978, 6.71097907961502948186435297164, 6.91673632582524755794371406427, 7.34279883968137600047389740645, 7.83413391287792649953413884502
