| L(s) = 1 | + (−1.53e11 − 2.65e11i)2-s + (−5.82e17 − 5.18e17i)3-s + (−2.81e22 + 4.88e22i)4-s + (1.84e25 − 3.18e25i)5-s + (−4.82e28 + 2.34e29i)6-s + (2.00e31 + 3.47e31i)7-s + (5.69e33 − 1.15e18i)8-s + (7.14e34 + 6.04e35i)9-s − 1.12e37·10-s + (−9.28e38 − 1.60e39i)11-s + (4.17e40 − 1.38e40i)12-s + (−2.92e41 + 5.06e41i)13-s + (6.15e42 − 1.06e43i)14-s + (−2.72e43 + 9.05e42i)15-s + (1.90e44 + 3.29e44i)16-s − 1.36e46·17-s + ⋯ |
| L(s) = 1 | + (−0.789 − 1.36i)2-s + (−0.747 − 0.664i)3-s + (−0.745 + 1.29i)4-s + (0.113 − 0.195i)5-s + (−0.318 + 1.54i)6-s + (0.408 + 0.707i)7-s + 0.776·8-s + (0.117 + 0.993i)9-s − 0.357·10-s + (−0.822 − 1.42i)11-s + (1.41 − 0.470i)12-s + (−0.493 + 0.854i)13-s + (0.644 − 1.11i)14-s + (−0.214 + 0.0713i)15-s + (0.133 + 0.230i)16-s − 0.987·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(0.4808318299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4808318299\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.82e17 + 5.18e17i)T \) |
| good | 2 | \( 1 + (1.53e11 + 2.65e11i)T + (-1.88e22 + 3.27e22i)T^{2} \) |
| 5 | \( 1 + (-1.84e25 + 3.18e25i)T + (-1.32e52 - 2.29e52i)T^{2} \) |
| 7 | \( 1 + (-2.00e31 - 3.47e31i)T + (-1.20e63 + 2.08e63i)T^{2} \) |
| 11 | \( 1 + (9.28e38 + 1.60e39i)T + (-6.35e77 + 1.10e78i)T^{2} \) |
| 13 | \( 1 + (2.92e41 - 5.06e41i)T + (-1.75e83 - 3.04e83i)T^{2} \) |
| 17 | \( 1 + 1.36e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.17e48T + 8.06e95T^{2} \) |
| 23 | \( 1 + (-7.49e50 + 1.29e51i)T + (-6.73e101 - 1.16e102i)T^{2} \) |
| 29 | \( 1 + (-4.26e54 - 7.39e54i)T + (-2.39e109 + 4.14e109i)T^{2} \) |
| 31 | \( 1 + (-1.17e54 + 2.03e54i)T + (-3.55e111 - 6.16e111i)T^{2} \) |
| 37 | \( 1 + (5.51e58 - 9.54e4i)T + 4.12e117T^{2} \) |
| 41 | \( 1 + (-2.99e60 + 5.19e60i)T + (-4.54e120 - 7.87e120i)T^{2} \) |
| 43 | \( 1 + (1.33e60 + 2.30e60i)T + (-1.61e122 + 2.80e122i)T^{2} \) |
| 47 | \( 1 + (1.04e62 + 1.80e62i)T + (-1.27e125 + 2.21e125i)T^{2} \) |
| 53 | \( 1 + (7.59e64 + 1.63e11i)T + 2.09e129T^{2} \) |
| 59 | \( 1 + (1.49e66 - 2.59e66i)T + (-3.25e132 - 5.64e132i)T^{2} \) |
| 61 | \( 1 + (1.55e66 + 2.68e66i)T + (-3.96e133 + 6.87e133i)T^{2} \) |
| 67 | \( 1 + (1.94e68 - 3.36e68i)T + (-4.51e136 - 7.81e136i)T^{2} \) |
| 71 | \( 1 + (-9.89e68 + 1.95e15i)T + 6.98e138T^{2} \) |
| 73 | \( 1 + (-8.23e69 - 2.26e16i)T + 5.61e139T^{2} \) |
| 79 | \( 1 + (4.85e70 + 8.40e70i)T + (-1.04e142 + 1.81e142i)T^{2} \) |
| 83 | \( 1 + (4.98e71 + 8.62e71i)T + (-4.26e143 + 7.38e143i)T^{2} \) |
| 89 | \( 1 + (2.15e73 + 8.63e18i)T + 1.60e146T^{2} \) |
| 97 | \( 1 + (3.42e73 + 5.93e73i)T + (-5.09e148 + 8.81e148i)T^{2} \) |
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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
−10.59358955226999146914301116550, −9.068988744899893560260842724736, −8.457721162670012520407675035064, −7.09226040699411542831437008326, −5.72015154200568548847220701923, −4.81093917461518061681133376888, −3.10869543777763661166682428100, −2.31798015140095063278443837252, −1.42794653414101162113410347858, −0.58186047020145566835868861857,
0.20444602507591343557414428598, 1.18685746528079672124646410244, 2.91951394894584057477250552545, 4.62017398600479971279280151183, 5.10425018000375818194495543241, 6.29472648943980535608806526047, 7.26439259174063728679127350981, 7.925840092647289208043707464228, 9.517528778285917245894728794530, 10.01513634552835900585896625745
