| L(s) = 1 | + (−0.117 + 1.72i)3-s + i·5-s − i·7-s + (−2.97 − 0.406i)9-s + (2.78 + 1.79i)11-s − 5.07i·13-s + (−1.72 − 0.117i)15-s + 2.63·17-s + 4.98i·19-s + (1.72 + 0.117i)21-s − 6.91i·23-s − 25-s + (1.05 − 5.08i)27-s − 3.99·29-s − 5.23·31-s + ⋯ |
| L(s) = 1 | + (−0.0679 + 0.997i)3-s + 0.447i·5-s − 0.377i·7-s + (−0.990 − 0.135i)9-s + (0.839 + 0.542i)11-s − 1.40i·13-s + (−0.446 − 0.0303i)15-s + 0.639·17-s + 1.14i·19-s + (0.377 + 0.0256i)21-s − 1.44i·23-s − 0.200·25-s + (0.202 − 0.979i)27-s − 0.741·29-s − 0.940·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01064389197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01064389197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.117 - 1.72i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-2.78 - 1.79i)T \) |
| good | 13 | \( 1 + 5.07iT - 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 - 4.98iT - 19T^{2} \) |
| 23 | \( 1 + 6.91iT - 23T^{2} \) |
| 29 | \( 1 + 3.99T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - 2.25iT - 43T^{2} \) |
| 47 | \( 1 - 8.00iT - 47T^{2} \) |
| 53 | \( 1 + 1.36iT - 53T^{2} \) |
| 59 | \( 1 + 4.62iT - 59T^{2} \) |
| 61 | \( 1 - 8.77iT - 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 1.77iT - 71T^{2} \) |
| 73 | \( 1 - 0.712iT - 73T^{2} \) |
| 79 | \( 1 + 2.13iT - 79T^{2} \) |
| 83 | \( 1 + 7.44T + 83T^{2} \) |
| 89 | \( 1 - 2.14iT - 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.768520995541893293935082149040, −8.153627771821351329681156236672, −7.36944042225834899602677386275, −6.54793530640880407877711911015, −5.73456105747679818727750555564, −5.14712792273461865478625268418, −4.15159884991809697481696686048, −3.56992866258009345493842744409, −2.86299209606851015743521816169, −1.54291099392807589969658451716,
0.00283638424627653485890911214, 1.43381828231922575654087375234, 1.92135453278107954209873880987, 3.21438503657246893957899846607, 3.92867784592424253985012590551, 5.20017811576624349877921861737, 5.56625403663178118531366163420, 6.62920992566257658920356311794, 6.98246401194838256068949368028, 7.76504091998718674949407232928
