L(s) = 1 | + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.172 − 0.349i)5-s + (−0.382 − 0.923i)8-s + (−0.130 + 0.991i)9-s + (−0.349 − 0.172i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (0.499 + 0.866i)18-s + (−0.382 + 0.0761i)20-s + (0.516 − 0.672i)25-s + (−1.40 − 0.184i)26-s + (1.08 − 1.63i)29-s + (−0.991 + 0.130i)32-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (−0.172 − 0.349i)5-s + (−0.382 − 0.923i)8-s + (−0.130 + 0.991i)9-s + (−0.349 − 0.172i)10-s + (−1 − i)13-s + (−0.866 − 0.499i)16-s + (−0.130 − 0.991i)17-s + (0.499 + 0.866i)18-s + (−0.382 + 0.0761i)20-s + (0.516 − 0.672i)25-s + (−1.40 − 0.184i)26-s + (1.08 − 1.63i)29-s + (−0.991 + 0.130i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.597878770\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597878770\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.793 + 0.608i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.130 + 0.991i)T \) |
good | 3 | \( 1 + (0.130 - 0.991i)T^{2} \) |
| 5 | \( 1 + (0.172 + 0.349i)T + (-0.608 + 0.793i)T^{2} \) |
| 11 | \( 1 + (0.793 - 0.608i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 23 | \( 1 + (-0.130 - 0.991i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.130 + 0.991i)T^{2} \) |
| 37 | \( 1 + (0.369 + 0.125i)T + (0.793 + 0.608i)T^{2} \) |
| 41 | \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.0999 - 0.758i)T + (-0.965 + 0.258i)T^{2} \) |
| 59 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 61 | \( 1 + (-0.0726 - 1.10i)T + (-0.991 + 0.130i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-1.95 - 0.128i)T + (0.991 + 0.130i)T^{2} \) |
| 79 | \( 1 + (0.130 + 0.991i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \) |
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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
−8.524422730702641663542616301847, −7.74022384678998835459222553100, −7.01537509500803589715153290496, −6.05324576089272658151132775361, −5.08417798631233202688146585136, −4.90090419620025783302205002395, −3.88986878734010345866199654006, −2.71837192152904084717396592673, −2.29037872298039133824961009705, −0.70554248323279188985764800329,
1.79740825494458635710468635121, 3.04481220644206120176696145026, 3.59322714893075713214949751247, 4.58479934821096858671206508653, 5.18119361821294785818307948635, 6.32732795963748916731657236888, 6.69428475236129541431292209959, 7.30241482090559166487325622365, 8.286999929864570740216122454400, 8.876810130670382402099424680817