| L(s) = 1 | + (−0.402 + 2.28i)2-s + (1.56 − 2.70i)3-s + (−3.15 − 1.14i)4-s + (0.564 + 3.19i)5-s + (5.54 + 4.65i)6-s + (0.171 − 0.297i)7-s + (1.57 − 2.73i)8-s + (−3.38 − 5.85i)9-s − 7.52·10-s + (−0.101 − 0.576i)11-s + (−8.04 + 6.75i)12-s + (−2.75 − 2.31i)13-s + (0.608 + 0.510i)14-s + (9.53 + 3.47i)15-s + (0.443 + 0.371i)16-s + (−1.75 − 3.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.284 + 1.61i)2-s + (0.901 − 1.56i)3-s + (−1.57 − 0.574i)4-s + (0.252 + 1.43i)5-s + (2.26 + 1.89i)6-s + (0.0648 − 0.112i)7-s + (0.557 − 0.965i)8-s + (−1.12 − 1.95i)9-s − 2.37·10-s + (−0.0306 − 0.173i)11-s + (−2.32 + 1.94i)12-s + (−0.764 − 0.641i)13-s + (0.162 + 0.136i)14-s + (2.46 + 0.896i)15-s + (0.110 + 0.0929i)16-s + (−0.425 − 0.736i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.825813 + 0.504311i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.825813 + 0.504311i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (3.40 + 7.83i)T \) |
| good | 2 | \( 1 + (0.402 - 2.28i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.56 + 2.70i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.564 - 3.19i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.171 + 0.297i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.101 + 0.576i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (2.75 + 2.31i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.75 + 3.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.374 - 2.12i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 6.07i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0931 + 0.528i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.14 + 1.51i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.910 - 5.16i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (5.60 - 4.70i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.77 + 3.07i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.90 + 5.79i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.497 + 2.82i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.14 - 5.15i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.906 + 0.761i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.36 + 1.14i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.90 - 16.4i)T + (-66.7 - 24.2i)T^{2} \) |
| 79 | \( 1 + (6.67 + 5.59i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + 6.29T + 83T^{2} \) |
| 89 | \( 1 + (-1.13 - 0.949i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (4.80 + 8.32i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.77076651233533379805682183598, −13.95611811290242102330111985752, −13.36027356928424920278961343061, −11.72523422640940150720536937492, −9.843329004890461429323322546911, −8.463837187732157067533822944886, −7.41831595706202454990338148206, −6.98533244542804554949304290195, −5.87084777239680834573651948587, −2.78867858605486080646802193754,
2.37826372254447513367832523207, 4.12884871685591523247132768055, 4.86023508623703565260500233481, 8.563776472394205585935946649919, 9.021595144680910529388577361692, 9.903118761657719455132677102089, 10.79502769491689615333409227887, 12.13813632984733676468168231970, 13.10874035177358681775200038415, 14.24683905316404599355162500242
