| L(s) = 1 | + (−22.7 + 40.8i)3-s + (3.05 + 5.29i)5-s + (−422. + 731. i)7-s + (−1.15e3 − 1.85e3i)9-s + (−1.78e3 + 3.09e3i)11-s + (508. + 880. i)13-s + (−285. + 4.62i)15-s − 1.17e4·17-s + 5.02e3·19-s + (−2.03e4 − 3.38e4i)21-s + (−1.23e4 − 2.14e4i)23-s + (3.90e4 − 6.76e4i)25-s + (1.02e5 − 4.95e3i)27-s + (9.83e4 − 1.70e5i)29-s + (−5.90e4 − 1.02e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.485 + 0.873i)3-s + (0.0109 + 0.0189i)5-s + (−0.465 + 0.806i)7-s + (−0.527 − 0.849i)9-s + (−0.404 + 0.700i)11-s + (0.0641 + 0.111i)13-s + (−0.0218 + 0.000353i)15-s − 0.579·17-s + 0.168·19-s + (−0.478 − 0.798i)21-s + (−0.211 − 0.366i)23-s + (0.499 − 0.865i)25-s + (0.998 − 0.0484i)27-s + (0.748 − 1.29i)29-s + (−0.355 − 0.616i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.138667 - 0.159933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.138667 - 0.159933i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (22.7 - 40.8i)T \) |
| 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
−12.63169551988544274256009606123, −11.79789254493637387932249562895, −10.52992376500556040488584786506, −9.612599269988183790373117037673, −8.519363796484560259165447406969, −6.67016483696715240032550507417, −5.47770081775698984271883702375, −4.21132743089970089241219390358, −2.54076647605670676086817826036, −0.081602934712644762707555344299,
1.29733517198378956312505836891, 3.18605081622563419291268666369, 5.12186846540493627939398710189, 6.48268718677255417013445085697, 7.44187921661860174922653350427, 8.741187245883292618200756862958, 10.41566370286459617832913780415, 11.24358273586820026590035888607, 12.55313041218223479187194123576, 13.37742936180959078745378572478
