L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (0.923 − 0.382i)6-s + (−0.382 − 0.923i)8-s − 1.00i·9-s + 10-s + 0.765·11-s − 12-s + (0.707 + 0.707i)13-s + (0.382 − 0.923i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (−0.923 − 0.382i)20-s + ⋯ |
L(s) = 1 | + (−0.923 − 0.382i)2-s + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)4-s + (−0.923 + 0.382i)5-s + (0.923 − 0.382i)6-s + (−0.382 − 0.923i)8-s − 1.00i·9-s + 10-s + 0.765·11-s − 12-s + (0.707 + 0.707i)13-s + (0.382 − 0.923i)15-s + i·16-s + (−0.382 + 0.923i)18-s + (−0.923 − 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4635681579\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4635681579\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 + 0.382i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - 0.765T + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 1.84iT - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 0.765iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (-0.541 + 0.541i)T - iT^{2} \) |
| 89 | \( 1 - 0.765iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.831229095646386873611575196471, −9.083971228940529069802516785826, −8.447292860785482867895292384616, −7.43108267750955017507633309688, −6.65479375352587032998263648537, −6.01115235042142781681648512093, −4.44891617522519947386618359506, −3.87742855825956738592218482077, −2.93842527631062362337481380029, −1.21099946057303335924408527341,
0.65481148062080092977819628280, 1.78325175309551369784117433868, 3.35543705022065715252031079188, 4.69970521276086416415775279911, 5.61660869180286212613632396187, 6.43085006613359868772408041788, 7.12421390204641341548780380375, 7.945538988416037099638497265292, 8.408834529721026868192958282043, 9.293942454653276855434647300221