| L(s) = 1 | + 2·2-s − 4-s − 4·7-s − 8·8-s − 9-s + 4·13-s − 8·14-s − 7·16-s − 2·18-s + 8·26-s + 4·28-s − 4·29-s + 14·32-s + 36-s − 16·37-s − 8·47-s − 2·49-s − 4·52-s + 32·56-s − 8·58-s + 20·61-s + 4·63-s + 35·64-s − 12·67-s + 8·72-s − 32·73-s − 32·74-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 1.51·7-s − 2.82·8-s − 1/3·9-s + 1.10·13-s − 2.13·14-s − 7/4·16-s − 0.471·18-s + 1.56·26-s + 0.755·28-s − 0.742·29-s + 2.47·32-s + 1/6·36-s − 2.63·37-s − 1.16·47-s − 2/7·49-s − 0.554·52-s + 4.27·56-s − 1.05·58-s + 2.56·61-s + 0.503·63-s + 35/8·64-s − 1.46·67-s + 0.942·72-s − 3.74·73-s − 3.71·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7538377455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7538377455\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−10.11252471256851549071493146725, −9.880414091424418658132861013652, −9.319622416648071449324570591266, −8.898793165334234014658190533377, −8.650863611370196622799488715819, −8.439565831854081273674696441522, −7.64542286320086236267891934574, −7.03803684755553610062481005220, −6.58142517114378503756498125900, −6.08046582004109741827642485953, −5.97268090022861388670995459439, −5.32656410484753359525529312513, −5.05134077588186481344250961037, −4.43652580368320604926417707024, −3.84349923164529488241366663297, −3.53861214437567002509012578611, −3.21847617840072021611031640267, −2.74384908291398315982345197273, −1.61274626009101443027035383670, −0.32151865095394270729282004528,
0.32151865095394270729282004528, 1.61274626009101443027035383670, 2.74384908291398315982345197273, 3.21847617840072021611031640267, 3.53861214437567002509012578611, 3.84349923164529488241366663297, 4.43652580368320604926417707024, 5.05134077588186481344250961037, 5.32656410484753359525529312513, 5.97268090022861388670995459439, 6.08046582004109741827642485953, 6.58142517114378503756498125900, 7.03803684755553610062481005220, 7.64542286320086236267891934574, 8.439565831854081273674696441522, 8.650863611370196622799488715819, 8.898793165334234014658190533377, 9.319622416648071449324570591266, 9.880414091424418658132861013652, 10.11252471256851549071493146725
