| L(s) = 1 | + (−11.0 + 11.0i)2-s + (−99.2 − 99.2i)3-s + 269i·4-s + 2.18e3·6-s + (−8.60e3 − 8.60e3i)8-s + 1.96e4i·9-s + (2.66e4 − 2.66e4i)12-s + 5.20e4·16-s + (−2.10e5 + 2.10e5i)17-s + (−2.16e5 − 2.16e5i)18-s + 1.03e6i·19-s + (3.47e5 + 3.47e5i)23-s + 1.70e6i·24-s + (1.95e6 − 1.95e6i)27-s − 8.24e6·31-s + (3.83e6 − 3.83e6i)32-s + ⋯ |
| L(s) = 1 | + (−0.487 + 0.487i)2-s + (−0.707 − 0.707i)3-s + 0.525i·4-s + 0.688·6-s + (−0.743 − 0.743i)8-s + 0.999i·9-s + (0.371 − 0.371i)12-s + 0.198·16-s + (−0.610 + 0.610i)17-s + (−0.487 − 0.487i)18-s + 1.82i·19-s + (0.258 + 0.258i)23-s + 1.05i·24-s + (0.707 − 0.707i)27-s − 1.60·31-s + (0.646 − 0.646i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.286649 - 0.226859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.286649 - 0.226859i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (99.2 + 99.2i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (11.0 - 11.0i)T - 512iT^{2} \) |
| 7 | \( 1 + 4.03e7iT^{2} \) |
| 11 | \( 1 - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.06e10iT^{2} \) |
| 17 | \( 1 + (2.10e5 - 2.10e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 1.03e6iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-3.47e5 - 3.47e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.24e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e14iT^{2} \) |
| 41 | \( 1 - 3.27e14T^{2} \) |
| 43 | \( 1 - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-4.70e7 + 4.70e7i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (7.68e7 + 7.68e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.97e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.72e16iT^{2} \) |
| 71 | \( 1 - 4.58e16T^{2} \) |
| 73 | \( 1 - 5.88e16iT^{2} \) |
| 79 | \( 1 + 4.21e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (6.04e8 + 6.04e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.60e17iT^{2} \) |
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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
−12.47264105943355785026381159886, −11.48435166806075265291743601940, −10.19112616587677030582405864176, −8.673743464693534582471495236976, −7.70781653387246311363829553864, −6.72446625916902976959110198835, −5.60494368292015842542879028345, −3.75745047490238042825346750396, −1.84979377758533418799793945521, −0.17592590938500470586699811320,
0.907401606410959924206216905170, 2.68136318527357079034212681428, 4.53149901559091257778624416918, 5.60816764105624014732329760696, 6.90173412151997834106510944796, 8.962019323007152455700159738771, 9.557617352454570793340906180519, 10.92938191439454346630452691128, 11.20865747669544452488404430476, 12.58935229595714695433158665640
