| L(s) = 1 | + 6·3-s + 7·4-s + 27·9-s + 42·12-s − 15·16-s + 222·17-s + 12·23-s + 169·25-s + 108·27-s − 210·29-s + 189·36-s + 1.02e3·43-s − 90·48-s + 682·49-s + 1.33e3·51-s + 1.27e3·53-s + 466·61-s − 553·64-s + 1.55e3·68-s + 72·69-s + 1.01e3·75-s − 2.64e3·79-s + 405·81-s − 1.26e3·87-s + 84·92-s + 1.18e3·100-s + 714·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 7/8·4-s + 9-s + 1.01·12-s − 0.234·16-s + 3.16·17-s + 0.108·23-s + 1.35·25-s + 0.769·27-s − 1.34·29-s + 7/8·36-s + 3.64·43-s − 0.270·48-s + 1.98·49-s + 3.65·51-s + 3.31·53-s + 0.978·61-s − 1.08·64-s + 2.77·68-s + 0.125·69-s + 1.56·75-s − 3.77·79-s + 5/9·81-s − 1.55·87-s + 0.0951·92-s + 1.18·100-s + 0.703·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.995897731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.995897731\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 7 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 169 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 1762 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 111 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11602 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 105 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 49582 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 101017 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 84481 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 514 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 181402 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 639 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 50758 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 233 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 255950 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 149078 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 714025 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1324 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 487474 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1161934 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
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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
−10.49278502485980065516205231771, −10.27678623868498215438542083979, −9.952875142639320099087530157564, −9.271044832754378402893121260786, −8.896321945762358806975591332684, −8.636838522435431763154809929807, −7.82018965672483181596001548996, −7.57204882251368678362122128521, −7.13617338298115841646644137761, −7.02184628756134408776418005813, −5.87141184929533907082343479485, −5.73470371751573264762651872308, −5.24411001507637250431035731266, −4.17304356705798184224953442369, −3.92352572845401671204318336114, −3.23764905901582361463887967081, −2.56718115602339167860997700036, −2.42109905357614104623334134488, −1.25107733736386298459950918832, −0.942553052629613768336905339397,
0.942553052629613768336905339397, 1.25107733736386298459950918832, 2.42109905357614104623334134488, 2.56718115602339167860997700036, 3.23764905901582361463887967081, 3.92352572845401671204318336114, 4.17304356705798184224953442369, 5.24411001507637250431035731266, 5.73470371751573264762651872308, 5.87141184929533907082343479485, 7.02184628756134408776418005813, 7.13617338298115841646644137761, 7.57204882251368678362122128521, 7.82018965672483181596001548996, 8.636838522435431763154809929807, 8.896321945762358806975591332684, 9.271044832754378402893121260786, 9.952875142639320099087530157564, 10.27678623868498215438542083979, 10.49278502485980065516205231771
