L(s) = 1 | + 17.6·5-s + 14.9i·7-s − 55.1i·11-s − 9.02i·13-s + 14.1i·17-s − 139.·19-s + 86.1·23-s + 186.·25-s − 3.64·29-s + 273. i·31-s + 264. i·35-s + 405. i·37-s − 344. i·41-s − 241.·43-s + 297.·47-s + ⋯ |
L(s) = 1 | + 1.57·5-s + 0.808i·7-s − 1.51i·11-s − 0.192i·13-s + 0.202i·17-s − 1.68·19-s + 0.781·23-s + 1.49·25-s − 0.0233·29-s + 1.58i·31-s + 1.27i·35-s + 1.80i·37-s − 1.31i·41-s − 0.857·43-s + 0.924·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.127559925\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.127559925\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 17.6T + 125T^{2} \) |
| 7 | \( 1 - 14.9iT - 343T^{2} \) |
| 11 | \( 1 + 55.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 9.02iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 86.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 3.64T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 405. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 344. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 241.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 297.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 630. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 237. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 778.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 127. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 301. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 338. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
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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
−8.692607555216358230027938171932, −8.308006944443699571253941590163, −6.72621382542184750808513841592, −6.35498109908465326657943003224, −5.50649792701533840111967599076, −5.07993423654529996369880988804, −3.62020941373052540401107445126, −2.67066028340943415566112980387, −1.95205854414461185377561041648, −0.824680064684913244358247700850,
0.76016858228777389169821582588, 2.02210412568439301271509356632, 2.34223199584481936955267150081, 3.94490841111413473044266702448, 4.62794186354551627252975414305, 5.52010627095875909432907388365, 6.37979249756009632634433428389, 6.97246930308777686235096094706, 7.71754297351202708560362109805, 8.890464612817617581064628672144