L(s) = 1 | + (−4.13 − 3.15i)3-s + 1.18i·5-s − 7i·7-s + (7.12 + 26.0i)9-s + 64.7·11-s − 65.6·13-s + (3.73 − 4.88i)15-s + 19.9i·17-s + 67.2i·19-s + (−22.0 + 28.9i)21-s + 79.6·23-s + 123.·25-s + (52.7 − 130. i)27-s − 100. i·29-s − 278. i·31-s + ⋯ |
L(s) = 1 | + (−0.794 − 0.606i)3-s + 0.105i·5-s − 0.377i·7-s + (0.263 + 0.964i)9-s + 1.77·11-s − 1.39·13-s + (0.0642 − 0.0841i)15-s + 0.285i·17-s + 0.812i·19-s + (−0.229 + 0.300i)21-s + 0.722·23-s + 0.988·25-s + (0.375 − 0.926i)27-s − 0.643i·29-s − 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.353920270\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353920270\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.13 + 3.15i)T \) |
| 7 | \( 1 + 7iT \) |
good | 5 | \( 1 - 1.18iT - 125T^{2} \) |
| 11 | \( 1 - 64.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 19.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 67.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 79.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 278. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 45.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 12.4iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 368. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 303.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 639. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 533. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 517. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3T + 9.12e5T^{2} \) |
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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
−11.10685375147153886284999796417, −10.12273285747937596224000404815, −9.194992896004497457119128761102, −7.84841808066070240775226989354, −6.97511776689817699003185261491, −6.25323276335801450369507225849, −5.02567499367967096384392996466, −3.89020008288595356641319552168, −2.05002092981949777216845353086, −0.67668771527139311584086867278,
1.08119841480815437270617945960, 3.05226484963829954436603377400, 4.47470755047209646677903810585, 5.17721183407597820191721318267, 6.50108971479181790930954751462, 7.14249719867140954083820986410, 8.993352603245530768794241632501, 9.277565009231947955572859702656, 10.44451776919297283625841410938, 11.34843619721212758663288398155