L(s) = 1 | + (0.259 + 0.259i)3-s + (1.45 + 1.45i)5-s + 0.413i·7-s − 2.86i·9-s + (−3.30 + 0.217i)11-s + (1.40 + 1.40i)13-s + 0.754i·15-s + 7.18i·17-s + (−4.93 + 4.93i)19-s + (−0.107 + 0.107i)21-s + 6.32·23-s − 0.764i·25-s + (1.51 − 1.51i)27-s + (−0.338 − 0.338i)29-s + 6.19i·31-s + ⋯ |
L(s) = 1 | + (0.149 + 0.149i)3-s + (0.650 + 0.650i)5-s + 0.156i·7-s − 0.955i·9-s + (−0.997 + 0.0654i)11-s + (0.390 + 0.390i)13-s + 0.194i·15-s + 1.74i·17-s + (−1.13 + 1.13i)19-s + (−0.0233 + 0.0233i)21-s + 1.31·23-s − 0.152i·25-s + (0.292 − 0.292i)27-s + (−0.0628 − 0.0628i)29-s + 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0359 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.668475112\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668475112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + (3.30 - 0.217i)T \) |
good | 3 | \( 1 + (-0.259 - 0.259i)T + 3iT^{2} \) |
| 5 | \( 1 + (-1.45 - 1.45i)T + 5iT^{2} \) |
| 7 | \( 1 - 0.413iT - 7T^{2} \) |
| 13 | \( 1 + (-1.40 - 1.40i)T + 13iT^{2} \) |
| 17 | \( 1 - 7.18iT - 17T^{2} \) |
| 19 | \( 1 + (4.93 - 4.93i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.32T + 23T^{2} \) |
| 29 | \( 1 + (0.338 + 0.338i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.19iT - 31T^{2} \) |
| 37 | \( 1 + (-6.64 - 6.64i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.706T + 41T^{2} \) |
| 43 | \( 1 + (2.33 + 2.33i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.37iT - 47T^{2} \) |
| 53 | \( 1 + (4.01 + 4.01i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.93 - 2.93i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.52 - 6.52i)T + 61iT^{2} \) |
| 67 | \( 1 + (2.43 + 2.43i)T + 67iT^{2} \) |
| 71 | \( 1 + 7.09T + 71T^{2} \) |
| 73 | \( 1 + 7.56T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + (-7.25 + 7.25i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.53iT - 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.989681971848057955532575176876, −8.800111364756502087944043952720, −8.396970946353879848357675506479, −7.23367333594329364956334167635, −6.20268823167827761900878569955, −6.01376920419100806767143475671, −4.62179096824612468197332665900, −3.62616925426503678703879037157, −2.70354811570765000941173287086, −1.54502756728941324080675792207,
0.66100684197230117071657674155, 2.20325422447901969375540858257, 2.90851778991538810033734900177, 4.56717601641249496614540670608, 5.10393886499697680687870365274, 5.88107061021936160863138067574, 7.13334392447800825065013478785, 7.67408455781314499378039275224, 8.691814895855421036561064434526, 9.225763416016108715087789491324