| L(s) = 1 | − 1.08·5-s + (2.10 − 1.60i)7-s − 1.23i·11-s − 3.69i·13-s − 6.30·17-s − 7.76i·19-s + 7.17i·23-s − 3.82·25-s − 1.41i·29-s + 4.54i·31-s + (−2.27 + 1.74i)35-s − 2.82·37-s + 9.37·41-s − 7.68·43-s − 10.9·47-s + ⋯ |
| L(s) = 1 | − 0.484·5-s + (0.794 − 0.607i)7-s − 0.371i·11-s − 1.02i·13-s − 1.53·17-s − 1.78i·19-s + 1.49i·23-s − 0.765·25-s − 0.262i·29-s + 0.816i·31-s + (−0.384 + 0.294i)35-s − 0.464·37-s + 1.46·41-s − 1.17·43-s − 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2534047012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2534047012\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.10 + 1.60i)T \) |
| good | 5 | \( 1 + 1.08T + 5T^{2} \) |
| 11 | \( 1 + 1.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.69iT - 13T^{2} \) |
| 17 | \( 1 + 6.30T + 17T^{2} \) |
| 19 | \( 1 + 7.76iT - 19T^{2} \) |
| 23 | \( 1 - 7.17iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 4.54iT - 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 9.37T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 5.41iT - 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 - 9.55iT - 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 10.6iT - 71T^{2} \) |
| 73 | \( 1 - 4.59iT - 73T^{2} \) |
| 79 | \( 1 - 9.42T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 - 5.86iT - 97T^{2} \) |
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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
−8.033277954074739079168584591638, −7.32882283438904248796588510655, −6.78418436300189882893966669510, −5.71396231402597503412523694980, −4.94385331528282800892154137079, −4.29051299811397561487957087665, −3.41548193053392174154257147525, −2.47355367614973228694175407099, −1.25598125109871401250433459031, −0.07176553385007682073929494212,
1.75801285100904489653997109438, 2.23705101174546809950795203945, 3.60821894653269444924224939633, 4.41125323223272216174043306526, 4.88011970507776407464331492903, 6.05310195526044741460874369661, 6.53692238596390951263278465479, 7.51186088934686112166284040581, 8.182752524503937908151447057766, 8.689010144986420453583508109029
