| L(s) = 1 | + 1.41i·3-s + 2.23i·5-s + (−1.58 − 2.12i)7-s + 0.999·9-s − 3.16·15-s + (3 − 2.23i)21-s − 9.48·23-s − 5.00·25-s + 5.65i·27-s − 6·29-s + (4.74 − 3.53i)35-s + 4.47i·41-s − 3.16·43-s + 2.23i·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s + 0.999i·5-s + (−0.597 − 0.801i)7-s + 0.333·9-s − 0.816·15-s + (0.654 − 0.487i)21-s − 1.97·23-s − 1.00·25-s + 1.08i·27-s − 1.11·29-s + (0.801 − 0.597i)35-s + 0.698i·41-s − 0.482·43-s + 0.333i·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3236200400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3236200400\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (1.58 + 2.12i)T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 9.48T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.836220317065446624383828226611, −8.907695826236335353906990318144, −7.74782210935106997026078220452, −7.23454530758490929662777948829, −6.39604913966852213513657335430, −5.65901311356172400429567334463, −4.39527730620454341089462879348, −3.84229296183604059600093067715, −3.11321661939200672856707800224, −1.82344890617754623162590092547,
0.10538841572877279844088412876, 1.54223441558543486333070527986, 2.30184739057377166911487198745, 3.66395273898151157003227522023, 4.54024004196710036213734526959, 5.63273661593517128024595602815, 6.11376549452673924159498369423, 7.04024146440145042179664450719, 7.943689019230533430428066463326, 8.388447596829142251982290470797
