| L(s) = 1 | − 6·3-s − 8·5-s + 4·7-s + 27·9-s + 40·11-s + 36·13-s + 48·15-s − 128·17-s − 196·19-s − 24·21-s + 46·23-s − 74·25-s − 108·27-s − 76·29-s − 168·31-s − 240·33-s − 32·35-s − 28·37-s − 216·39-s + 292·41-s + 780·43-s − 216·45-s + 432·47-s − 26·49-s + 768·51-s − 368·53-s − 320·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.715·5-s + 0.215·7-s + 9-s + 1.09·11-s + 0.768·13-s + 0.826·15-s − 1.82·17-s − 2.36·19-s − 0.249·21-s + 0.417·23-s − 0.591·25-s − 0.769·27-s − 0.486·29-s − 0.973·31-s − 1.26·33-s − 0.154·35-s − 0.124·37-s − 0.886·39-s + 1.11·41-s + 2.76·43-s − 0.715·45-s + 1.34·47-s − 0.0758·49-s + 2.10·51-s − 0.953·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5777730152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5777730152\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_4$ | \( 1 - 4 T + 6 p T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 40 T + 2670 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 36 T + 3150 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 128 T + 11034 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 196 T + 21274 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 76 T + 23310 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 168 T + 62766 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 28 T + 95670 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 292 T + 151958 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 780 T + 299562 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 432 T + 241502 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 368 T + 42810 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 40 T + 346358 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 556 T + 478758 T^{2} + 556 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 460 T + 571194 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2112 T + 1821710 T^{2} - 2112 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1548 T + 1364310 T^{2} + 1548 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1420 T + 1017786 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 216 T + 1119326 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1112 T + 1581786 T^{2} + 1112 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 348 T + 1425030 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.72333239338764522635719584312, −10.68831662653490366372437614285, −9.439676352284864840604818810118, −9.430540420609958118230234555437, −8.679654681053926374106984607093, −8.676097764771118105162416112154, −7.72944715710183126585735769360, −7.49477767896691336373836698121, −6.79062377395671677757698042800, −6.51673604665231398389287440982, −5.93958329822060651267944572901, −5.83756694330983255693371669468, −4.78238024789666844304190774673, −4.44439475525354240040167091830, −3.88199459051288060782232585828, −3.86059166976110884683220124098, −2.45980045672589764985553401458, −1.96762990670601376326323064255, −1.11257299332890150261943493110, −0.27931873183518841282782309369,
0.27931873183518841282782309369, 1.11257299332890150261943493110, 1.96762990670601376326323064255, 2.45980045672589764985553401458, 3.86059166976110884683220124098, 3.88199459051288060782232585828, 4.44439475525354240040167091830, 4.78238024789666844304190774673, 5.83756694330983255693371669468, 5.93958329822060651267944572901, 6.51673604665231398389287440982, 6.79062377395671677757698042800, 7.49477767896691336373836698121, 7.72944715710183126585735769360, 8.676097764771118105162416112154, 8.679654681053926374106984607093, 9.430540420609958118230234555437, 9.439676352284864840604818810118, 10.68831662653490366372437614285, 10.72333239338764522635719584312
