| L(s) = 1 | + 4·5-s − 2·7-s − 4·11-s + 2·13-s − 2·17-s − 2·19-s + 6·23-s + 2·25-s − 10·29-s + 6·31-s − 8·35-s − 10·37-s + 8·43-s + 3·49-s − 2·53-s − 16·55-s + 4·59-s − 4·61-s + 8·65-s − 12·67-s − 6·71-s − 20·73-s + 8·77-s + 18·79-s + 10·83-s − 8·85-s − 16·89-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.458·19-s + 1.25·23-s + 2/5·25-s − 1.85·29-s + 1.07·31-s − 1.35·35-s − 1.64·37-s + 1.21·43-s + 3/7·49-s − 0.274·53-s − 2.15·55-s + 0.520·59-s − 0.512·61-s + 0.992·65-s − 1.46·67-s − 0.712·71-s − 2.34·73-s + 0.911·77-s + 2.02·79-s + 1.09·83-s − 0.867·85-s − 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_4$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 222 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52976781513683755530328151294, −7.12478070566499264673362598198, −6.64233612564993790734009632383, −6.62165980050771849222973935862, −5.94439462795339759171571187797, −5.88257156651207040240560299792, −5.49954923422036470323044211564, −5.38710787028660319984339311543, −4.73928513783613514319005326355, −4.54971332372512253874145288967, −3.83022111767345617794746055628, −3.73972561851802654551471421950, −2.94304353187355982524698371135, −2.88819445948934469215030515990, −2.26631660693678829317883843248, −2.15733105751719477165458408885, −1.35803360494428203413027884755, −1.30198454774280321651165281559, 0, 0,
1.30198454774280321651165281559, 1.35803360494428203413027884755, 2.15733105751719477165458408885, 2.26631660693678829317883843248, 2.88819445948934469215030515990, 2.94304353187355982524698371135, 3.73972561851802654551471421950, 3.83022111767345617794746055628, 4.54971332372512253874145288967, 4.73928513783613514319005326355, 5.38710787028660319984339311543, 5.49954923422036470323044211564, 5.88257156651207040240560299792, 5.94439462795339759171571187797, 6.62165980050771849222973935862, 6.64233612564993790734009632383, 7.12478070566499264673362598198, 7.52976781513683755530328151294
