| L(s) = 1 | + (−3.05 + 5.29i)5-s + (422. + 731. i)7-s + (−1.78e3 − 3.09e3i)11-s + (508. − 880. i)13-s + 1.17e4·17-s − 5.02e3·19-s + (−1.23e4 + 2.14e4i)23-s + (3.90e4 + 6.76e4i)25-s + (−9.83e4 − 1.70e5i)29-s + (5.90e4 − 1.02e5i)31-s − 5.16e3·35-s − 3.82e5·37-s + (1.95e5 − 3.39e5i)41-s + (−1.37e5 − 2.37e5i)43-s + (1.78e5 + 3.08e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.0109 + 0.0189i)5-s + (0.465 + 0.806i)7-s + (−0.404 − 0.700i)11-s + (0.0641 − 0.111i)13-s + 0.579·17-s − 0.168·19-s + (−0.211 + 0.366i)23-s + (0.499 + 0.865i)25-s + (−0.748 − 1.29i)29-s + (0.355 − 0.616i)31-s − 0.0203·35-s − 1.24·37-s + (0.443 − 0.768i)41-s + (−0.263 − 0.455i)43-s + (0.250 + 0.433i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.881574195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881574195\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (3.05 - 5.29i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-422. - 731. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.78e3 + 3.09e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-508. + 880. i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.02e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (1.23e4 - 2.14e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (9.83e4 + 1.70e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-5.90e4 + 1.02e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.95e5 + 3.39e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.37e5 + 2.37e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.78e5 - 3.08e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 - 7.97e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.27e5 - 5.67e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (8.57e4 + 1.48e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.30e6 - 2.25e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 3.94e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-3.76e6 - 6.52e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.65e5 + 6.33e5i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 7.16e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (6.99e6 + 1.21e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−9.830266266399089406145467247313, −8.857059863806744076758647080813, −8.129990651733709457069119102273, −7.19216649287992956885464058299, −5.82771148283210537405359430499, −5.35818301530322908089642809213, −3.98180225026575580706941144214, −2.85253858620579267230837341262, −1.78325493110937075893900648851, −0.44793809827729524640923244536,
0.884978700201472039063198010730, 1.95563314524399193417614968207, 3.28771703472344371450044855466, 4.42769769842691616872659819957, 5.20527991179500769238181986516, 6.54102314719849845441201686279, 7.38504348774222517491160231329, 8.195569139984950864776147108300, 9.219440275053289604715851833319, 10.36194256135034847378715815039
