| L(s) = 1 | + (−0.726 + 1.57i)3-s + (2.19 − 0.411i)5-s + (−1.94 − 2.28i)9-s + 1.72i·11-s + 1.24·13-s + (−0.951 + 3.75i)15-s + 4.75i·17-s − 4.23i·19-s − 2.60·23-s + (4.66 − 1.80i)25-s + (5.00 − 1.39i)27-s + 2.97i·29-s − 0.179i·31-s + (−2.71 − 1.25i)33-s + 5.41i·37-s + ⋯ |
| L(s) = 1 | + (−0.419 + 0.907i)3-s + (0.982 − 0.183i)5-s + (−0.647 − 0.761i)9-s + 0.520i·11-s + 0.344·13-s + (−0.245 + 0.969i)15-s + 1.15i·17-s − 0.971i·19-s − 0.543·23-s + (0.932 − 0.361i)25-s + (0.963 − 0.268i)27-s + 0.552i·29-s − 0.0322i·31-s + (−0.472 − 0.218i)33-s + 0.890i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684236065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684236065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.726 - 1.57i)T \) |
| 5 | \( 1 + (-2.19 + 0.411i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 1.72iT - 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 4.75iT - 17T^{2} \) |
| 19 | \( 1 + 4.23iT - 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 2.97iT - 29T^{2} \) |
| 31 | \( 1 + 0.179iT - 31T^{2} \) |
| 37 | \( 1 - 5.41iT - 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 8.03iT - 43T^{2} \) |
| 47 | \( 1 - 7.87iT - 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 - 2.07T + 59T^{2} \) |
| 61 | \( 1 + 9.34iT - 61T^{2} \) |
| 67 | \( 1 - 9.46iT - 67T^{2} \) |
| 71 | \( 1 - 8.56iT - 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 0.643T + 79T^{2} \) |
| 83 | \( 1 + 4.73iT - 83T^{2} \) |
| 89 | \( 1 - 8.25T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.934994789200192770762057230635, −8.591234375152413497090526334723, −7.38884954044478253869252777676, −6.30407041427257302880608458310, −6.02062503253888657079881525998, −4.99424611215892309435269743433, −4.47207610460542154589916965490, −3.44407129978116208111378998851, −2.41169897012424740600039109865, −1.20327733287964630818564382183,
0.60184003112781034886479722102, 1.78859097445759338543841156676, 2.54240823111235867778907461339, 3.62758415286897282952409270638, 4.94524800859443424911365638485, 5.75846637118913591017842834910, 6.09766545377349774828614155160, 7.03233048558560673327873879385, 7.61354475873703298143374708901, 8.568605116229358232364202459595
