L(s) = 1 | + 2.51i·2-s + 2.41·3-s − 4.34·4-s + 2.06·5-s + 6.08i·6-s + 1.33i·7-s − 5.89i·8-s + 2.83·9-s + 5.20i·10-s + 1.77i·11-s − 10.4·12-s − 4.54i·13-s − 3.37·14-s + 4.99·15-s + 6.16·16-s + 6.51·17-s + ⋯ |
L(s) = 1 | + 1.78i·2-s + 1.39·3-s − 2.17·4-s + 0.923·5-s + 2.48i·6-s + 0.506i·7-s − 2.08i·8-s + 0.946·9-s + 1.64i·10-s + 0.535i·11-s − 3.02·12-s − 1.25i·13-s − 0.901·14-s + 1.28·15-s + 1.54·16-s + 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.772355 + 1.87252i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.772355 + 1.87252i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 + (-13.2 - 13.1i)T \) |
good | 2 | \( 1 - 2.51iT - 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 - 1.33iT - 7T^{2} \) |
| 11 | \( 1 - 1.77iT - 11T^{2} \) |
| 13 | \( 1 + 4.54iT - 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 19 | \( 1 + 7.86T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 + 0.268T + 31T^{2} \) |
| 37 | \( 1 - 4.96T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 1.04iT - 47T^{2} \) |
| 53 | \( 1 + 6.53iT - 53T^{2} \) |
| 59 | \( 1 + 6.20iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 - 8.05iT - 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 + 3.06iT - 89T^{2} \) |
| 97 | \( 1 + 0.353iT - 97T^{2} \) |
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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
−12.42915857050129511004947588334, −10.20005021028819893456201098764, −9.658385773812862003048614951856, −8.587248625854219776218699743028, −8.181443046132960508070712222602, −7.25304811901831467566499052848, −6.03368106240699484879018944054, −5.36616478103732466900929635808, −3.89481379882259969687098541658, −2.35444917185468319826915986303,
1.58601649113635953082782565581, 2.48099534094293588841021499833, 3.57052091314212794577232328923, 4.44847877018711060393585612796, 6.16842709770605974005647556722, 7.891894686099856216068959710243, 8.780202077030517495913862264442, 9.543704689998299648288648145740, 10.09924637677030330422530541147, 11.01125418204191791997539091397