| L(s) = 1 | − 5·3-s − 9·5-s − 28·7-s + 27·9-s − 57·11-s + 140·13-s + 45·15-s − 51·17-s + 5·19-s + 140·21-s − 69·23-s + 125·25-s − 280·27-s − 228·29-s − 23·31-s + 285·33-s + 252·35-s − 253·37-s − 700·39-s − 84·41-s + 248·43-s − 243·45-s − 201·47-s + 441·49-s + 255·51-s − 393·53-s + 513·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s − 0.804·5-s − 1.51·7-s + 9-s − 1.56·11-s + 2.98·13-s + 0.774·15-s − 0.727·17-s + 0.0603·19-s + 1.45·21-s − 0.625·23-s + 25-s − 1.99·27-s − 1.45·29-s − 0.133·31-s + 1.50·33-s + 1.21·35-s − 1.12·37-s − 2.87·39-s − 0.319·41-s + 0.879·43-s − 0.804·45-s − 0.623·47-s + 9/7·49-s + 0.700·51-s − 1.01·53-s + 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 5 T - 2 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 9 T - 44 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 57 T + 1918 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 p T - 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T - 6834 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 p T - 14 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 114 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 23 T - 29262 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 253 T + 13356 T^{2} + 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 201 T - 63422 T^{2} + 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 393 T + 5572 T^{2} + 393 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 219 T - 157418 T^{2} - 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 709 T + 275700 T^{2} + 709 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 419 T - 125202 T^{2} - 419 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 313 T - 291048 T^{2} - 313 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 461 T - 280518 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 588 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1017 T + 329320 T^{2} - 1017 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1834 T + p^{3} T^{2} )^{2} \) |
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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
−10.45093906725372977515994202723, −10.32268203754238261811658589902, −9.554064299203757602277690623459, −9.109733863832044360344825303738, −8.733689343574112062571223348206, −8.002871006466721802766127625533, −7.78582931463278064391922138549, −7.13159965780730776550447186849, −6.45364502144832881346747165143, −6.40059570899197930507171424470, −5.63541482430401341330725914353, −5.47169597494719786697005321640, −4.57496237718233328623149648036, −3.80876302484631115657618471494, −3.68703376024002654347866858187, −3.05980822046880353870539023238, −2.00025111297635820819800713013, −1.13604582103349351145320375145, 0, 0,
1.13604582103349351145320375145, 2.00025111297635820819800713013, 3.05980822046880353870539023238, 3.68703376024002654347866858187, 3.80876302484631115657618471494, 4.57496237718233328623149648036, 5.47169597494719786697005321640, 5.63541482430401341330725914353, 6.40059570899197930507171424470, 6.45364502144832881346747165143, 7.13159965780730776550447186849, 7.78582931463278064391922138549, 8.002871006466721802766127625533, 8.733689343574112062571223348206, 9.109733863832044360344825303738, 9.554064299203757602277690623459, 10.32268203754238261811658589902, 10.45093906725372977515994202723
