L(s) = 1 | + (9.70 + 5.81i)2-s + (60.3 + 112. i)4-s − 324. i·5-s − 956.·7-s + (−70.1 + 1.44e3i)8-s + (1.88e3 − 3.14e3i)10-s − 5.45e3i·11-s − 6.28e3i·13-s + (−9.28e3 − 5.56e3i)14-s + (−9.09e3 + 1.36e4i)16-s − 3.45e4·17-s − 1.45e4i·19-s + (3.66e4 − 1.95e4i)20-s + (3.17e4 − 5.29e4i)22-s + 2.46e4·23-s + ⋯ |
L(s) = 1 | + (0.857 + 0.513i)2-s + (0.471 + 0.881i)4-s − 1.16i·5-s − 1.05·7-s + (−0.0484 + 0.998i)8-s + (0.596 − 0.995i)10-s − 1.23i·11-s − 0.793i·13-s + (−0.904 − 0.541i)14-s + (−0.554 + 0.831i)16-s − 1.70·17-s − 0.488i·19-s + (1.02 − 0.547i)20-s + (0.634 − 1.05i)22-s + 0.422·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0484 + 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0484 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.15702 - 1.21446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15702 - 1.21446i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-9.70 - 5.81i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 324. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 956.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.45e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 6.28e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.45e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.71e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.03e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 7.16e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.28e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.04e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.25e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 - 1.55e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.16e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.38e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.89e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 4.37e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.84e6T + 8.07e13T^{2} \) |
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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
−13.28777633384057606783632631489, −12.18331659167327056104803139836, −10.94267238325725734511494669486, −9.131361520225358944869146114252, −8.249835525526135810178408816272, −6.63268587082342441404846244904, −5.57776960157929975579076387267, −4.27014392347832649548468165581, −2.82442743394406087709862844087, −0.40397294169148143363687917825,
2.05307519545334511973140157664, 3.24962731139650811800349523121, 4.60761908092862012056191107797, 6.51850787555933153418523545073, 6.91495388914397436267461098013, 9.368990626694628900530027430509, 10.33139509632270565955506449903, 11.29453106634439176504928487295, 12.48107137445936482129495947049, 13.38954785617075056366465466215