| L(s) = 1 | + 256·4-s − 431·5-s − 2.40e3·7-s + 6.56e3·9-s − 2.85e4·13-s + 6.55e4·16-s + 2.51e5·19-s − 1.10e5·20-s + 3.75e5·23-s − 2.04e5·25-s − 6.14e5·28-s − 1.06e6·29-s + 6.30e5·31-s + 1.03e6·35-s + 1.67e6·36-s − 4.09e5·41-s + 6.65e6·43-s − 2.82e6·45-s + 5.16e6·47-s + 5.76e6·49-s − 7.31e6·52-s + 1.33e7·53-s + 2.29e7·59-s − 1.57e7·63-s + 1.67e7·64-s + 1.23e7·65-s − 3.95e7·73-s + ⋯ |
| L(s) = 1 | + 4-s − 0.689·5-s − 7-s + 9-s − 13-s + 16-s + 1.92·19-s − 0.689·20-s + 1.34·23-s − 0.524·25-s − 28-s − 1.50·29-s + 0.682·31-s + 0.689·35-s + 36-s − 0.145·41-s + 1.94·43-s − 0.689·45-s + 1.05·47-s + 49-s − 52-s + 1.68·53-s + 1.89·59-s − 63-s + 64-s + 0.689·65-s − 1.39·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.202161262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202161262\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{4} T \) |
| 13 | \( 1 + p^{4} T \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 3 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 5 | \( 1 + 431 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 - 251233 T + p^{8} T^{2} \) |
| 23 | \( 1 - 375407 T + p^{8} T^{2} \) |
| 29 | \( 1 + 1062913 T + p^{8} T^{2} \) |
| 31 | \( 1 - 630433 T + p^{8} T^{2} \) |
| 37 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 41 | \( 1 + 409922 T + p^{8} T^{2} \) |
| 43 | \( 1 - 6653327 T + p^{8} T^{2} \) |
| 47 | \( 1 - 5166913 T + p^{8} T^{2} \) |
| 53 | \( 1 - 13303487 T + p^{8} T^{2} \) |
| 59 | \( 1 - 22939678 T + p^{8} T^{2} \) |
| 61 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 67 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 + 39577007 T + p^{8} T^{2} \) |
| 79 | \( 1 + 14012113 T + p^{8} T^{2} \) |
| 83 | \( 1 + 37402367 T + p^{8} T^{2} \) |
| 89 | \( 1 + 4088207 T + p^{8} T^{2} \) |
| 97 | \( 1 - 174312913 T + p^{8} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.31537775761649252132434870378, −11.54896513405971253837390631262, −10.27438789189705580553115303707, −9.386378289406852433594706002212, −7.38458116143086261747259785992, −7.18918702351412774866563673639, −5.55960937461033205346154137753, −3.82530205555607426144551636715, −2.63090645296419318962016577961, −0.901905524650583925534898585636,
0.901905524650583925534898585636, 2.63090645296419318962016577961, 3.82530205555607426144551636715, 5.55960937461033205346154137753, 7.18918702351412774866563673639, 7.38458116143086261747259785992, 9.386378289406852433594706002212, 10.27438789189705580553115303707, 11.54896513405971253837390631262, 12.31537775761649252132434870378
