L(s) = 1 | + 1.73i·3-s + (1.77 + 1.35i)5-s − 3.32i·7-s − 0.0262·9-s + 5.77·11-s + 1.10i·13-s + (−2.36 + 3.09i)15-s − 0.893i·17-s − 2.42·19-s + 5.77·21-s + i·23-s + (1.31 + 4.82i)25-s + 5.17i·27-s − 4.11·29-s − 9.54·31-s + ⋯ |
L(s) = 1 | + 1.00i·3-s + (0.794 + 0.606i)5-s − 1.25i·7-s − 0.00874·9-s + 1.74·11-s + 0.305i·13-s + (−0.609 + 0.798i)15-s − 0.216i·17-s − 0.557·19-s + 1.26·21-s + 0.208i·23-s + (0.263 + 0.964i)25-s + 0.995i·27-s − 0.763·29-s − 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55714 + 0.770381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55714 + 0.770381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.77 - 1.35i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 7 | \( 1 + 3.32iT - 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 - 1.10iT - 13T^{2} \) |
| 17 | \( 1 + 0.893iT - 17T^{2} \) |
| 19 | \( 1 + 2.42T + 19T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 7.69iT - 37T^{2} \) |
| 41 | \( 1 - 0.00418T + 41T^{2} \) |
| 43 | \( 1 + 9.97iT - 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 6.25iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 1.89iT - 73T^{2} \) |
| 79 | \( 1 - 0.216T + 79T^{2} \) |
| 83 | \( 1 + 5.38iT - 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 + 2.08iT - 97T^{2} \) |
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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
−10.90925004903554939705209124487, −10.27979051917908206633326799769, −9.544226932274967399636789799781, −8.884431777214069200404198292444, −7.17575464916039918072371116289, −6.72555518775480411531548959974, −5.42135607989199443635511998964, −4.11108828328413617984339474937, −3.61782291742945433730528596659, −1.65228755905338632074656743298,
1.39302606939727175050758748720, 2.29695849510332593676984573594, 4.09180069317803433687768400450, 5.55665787961882555614448737647, 6.19554399045140400499820615893, 7.09464642642977371005937428900, 8.365063365204883289548467169816, 9.079046599855844559107792660169, 9.721008069749666763383995423310, 11.17235093795170441241591576446