| L(s) = 1 | − 2·3-s + 4-s + 3·9-s + 4·11-s − 2·12-s − 3·16-s − 4·17-s − 4·19-s − 8·23-s − 4·27-s − 4·29-s − 12·31-s − 8·33-s + 3·36-s − 4·37-s + 4·41-s + 4·44-s + 8·47-s + 6·48-s + 8·51-s + 16·53-s + 8·57-s + 4·61-s − 7·64-s + 8·67-s − 4·68-s + 16·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 9-s + 1.20·11-s − 0.577·12-s − 3/4·16-s − 0.970·17-s − 0.917·19-s − 1.66·23-s − 0.769·27-s − 0.742·29-s − 2.15·31-s − 1.39·33-s + 1/2·36-s − 0.657·37-s + 0.624·41-s + 0.603·44-s + 1.16·47-s + 0.866·48-s + 1.12·51-s + 2.19·53-s + 1.05·57-s + 0.512·61-s − 7/8·64-s + 0.977·67-s − 0.485·68-s + 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7318556147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7318556147\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.635960145945055757568345703296, −8.605084714092146135392632735867, −7.80054506978780456905077242337, −7.36242026990884799377344065406, −7.26658105961057605069252095006, −6.65477471878433815324520019001, −6.63427658909257226827743623018, −6.10608355086408323741169861026, −5.68567957709861055805565949735, −5.63503481568599403468685603418, −4.86982223580545757632643632193, −4.59342330779616719641612216256, −3.92832986459965507375876676913, −3.91022316466443389763169861341, −3.53301775422992055102375440473, −2.38835038602439435622537692844, −2.17083747928925924764881764933, −1.87263617156798994300816485130, −1.09965321179640228894749427548, −0.28790841549584961952345083584,
0.28790841549584961952345083584, 1.09965321179640228894749427548, 1.87263617156798994300816485130, 2.17083747928925924764881764933, 2.38835038602439435622537692844, 3.53301775422992055102375440473, 3.91022316466443389763169861341, 3.92832986459965507375876676913, 4.59342330779616719641612216256, 4.86982223580545757632643632193, 5.63503481568599403468685603418, 5.68567957709861055805565949735, 6.10608355086408323741169861026, 6.63427658909257226827743623018, 6.65477471878433815324520019001, 7.26658105961057605069252095006, 7.36242026990884799377344065406, 7.80054506978780456905077242337, 8.605084714092146135392632735867, 8.635960145945055757568345703296
