| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·7-s + 2·10-s − 7·13-s + 4·14-s − 4·16-s − 9·17-s − 4·19-s + 2·20-s + 12·23-s − 4·25-s − 14·26-s + 4·28-s − 5·29-s + 31-s − 8·32-s − 18·34-s + 2·35-s − 8·38-s − 13·43-s + 24·46-s − 3·49-s − 8·50-s − 14·52-s − 10·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.755·7-s + 0.632·10-s − 1.94·13-s + 1.06·14-s − 16-s − 2.18·17-s − 0.917·19-s + 0.447·20-s + 2.50·23-s − 4/5·25-s − 2.74·26-s + 0.755·28-s − 0.928·29-s + 0.179·31-s − 1.41·32-s − 3.08·34-s + 0.338·35-s − 1.29·38-s − 1.98·43-s + 3.53·46-s − 3/7·49-s − 1.13·50-s − 1.94·52-s − 1.31·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 5 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−8.675826706109194828227046842118, −7.88246088012404976606361098880, −7.25455976899288218444575798125, −6.96882691202060272557575805729, −6.55846848522447838952573009878, −6.05495636552206551800298917383, −5.33978532203042368017520388565, −4.93201962621514704485807356359, −4.73776836012580163127150603724, −4.28001634203246618188869523364, −3.51496730372918665400973531788, −2.80134164646229779790838476634, −2.25797904343745383163220514406, −1.83352154795621405101783657589, 0,
1.83352154795621405101783657589, 2.25797904343745383163220514406, 2.80134164646229779790838476634, 3.51496730372918665400973531788, 4.28001634203246618188869523364, 4.73776836012580163127150603724, 4.93201962621514704485807356359, 5.33978532203042368017520388565, 6.05495636552206551800298917383, 6.55846848522447838952573009878, 6.96882691202060272557575805729, 7.25455976899288218444575798125, 7.88246088012404976606361098880, 8.675826706109194828227046842118
