L(s) = 1 | − 3.46·5-s + (1 + 2.44i)7-s + 4.24i·11-s − 4.89i·13-s − 3.46·17-s − 4.89i·19-s − 4.24i·23-s + 6.99·25-s − 4.24i·29-s + (−3.46 − 8.48i)35-s − 8·37-s + 3.46·41-s + 2·43-s − 6.92·47-s + (−4.99 + 4.89i)49-s + ⋯ |
L(s) = 1 | − 1.54·5-s + (0.377 + 0.925i)7-s + 1.27i·11-s − 1.35i·13-s − 0.840·17-s − 1.12i·19-s − 0.884i·23-s + 1.39·25-s − 0.787i·29-s + (−0.585 − 1.43i)35-s − 1.31·37-s + 0.541·41-s + 0.304·43-s − 1.01·47-s + (−0.714 + 0.699i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4059086643\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4059086643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 4.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.604471380567261277684357706963, −8.661515213732875780563502436687, −8.041638768593873387583478589821, −7.32830048492726138541962209563, −6.40570421645634995482108107871, −5.01274627049784423938302269229, −4.54290578387824164368309994845, −3.30281762692379705279491977101, −2.23099435967430749903964436616, −0.19296566215816578712274547067,
1.41552142900784671795552119596, 3.32828193221980080036557981408, 3.97907493330040818301301053535, 4.73190106636080514521640132173, 6.10305942880796060122452535495, 7.10842077039591159762704676526, 7.70065033064568807562550561919, 8.511524194342010059850062677718, 9.202714827991693144751301953739, 10.58639687405641076287505089640