| L(s) = 1 | + (−1.25 + 0.648i)2-s + (1.15 − 1.63i)4-s − 1.27·5-s + i·7-s + (−0.396 + 2.80i)8-s + (1.60 − 0.828i)10-s + 6.57i·11-s − 4.05i·13-s + (−0.648 − 1.25i)14-s + (−1.31 − 3.77i)16-s − 6.09i·17-s − 2.08·19-s + (−1.47 + 2.08i)20-s + (−4.26 − 8.26i)22-s − 6.85·23-s + ⋯ |
| L(s) = 1 | + (−0.888 + 0.458i)2-s + (0.578 − 0.815i)4-s − 0.570·5-s + 0.377i·7-s + (−0.140 + 0.990i)8-s + (0.507 − 0.261i)10-s + 1.98i·11-s − 1.12i·13-s + (−0.173 − 0.335i)14-s + (−0.329 − 0.944i)16-s − 1.47i·17-s − 0.477·19-s + (−0.330 + 0.465i)20-s + (−0.909 − 1.76i)22-s − 1.42·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3167894792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3167894792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 - 0.648i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.27T + 5T^{2} \) |
| 11 | \( 1 - 6.57iT - 11T^{2} \) |
| 13 | \( 1 + 4.05iT - 13T^{2} \) |
| 17 | \( 1 + 6.09iT - 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 6.53T + 29T^{2} \) |
| 31 | \( 1 - 3.26iT - 31T^{2} \) |
| 37 | \( 1 + 2.95iT - 37T^{2} \) |
| 41 | \( 1 - 3.35iT - 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 7.75iT - 59T^{2} \) |
| 61 | \( 1 + 12.0iT - 61T^{2} \) |
| 67 | \( 1 + 3.01T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 + 0.849iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 + 4.06iT - 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−9.393095991442387473605819088291, −8.251684936341097341046270139236, −7.73939356160764276338326089862, −7.07915235939567264512681079703, −6.19343056480640736245231184741, −5.15126234254017086227429131901, −4.42022321157582411842297962675, −2.84612103683189906281892829054, −1.86319811430416850021876612338, −0.17917203480787117162117602310,
1.21725180509673937880650008003, 2.51557109610896399236196000509, 3.80322100972509194637271000614, 4.09244339021460077247390985514, 5.98928380501168004361263107575, 6.43813851124861679796642696549, 7.61941658605275658947370607738, 8.310957007901385308816100637437, 8.685501310562847644995137303843, 9.709650123563649156877224879110
