| L(s) = 1 | + i·2-s − 4-s + 3.09i·5-s + (−2.44 + i)7-s − i·8-s − 3.09·10-s + (1.79 + 2.79i)11-s − 0.646·13-s + (−1 − 2.44i)14-s + 16-s − 3.74·17-s + 1.80·19-s − 3.09i·20-s + (−2.79 + 1.79i)22-s − 4·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.38i·5-s + (−0.925 + 0.377i)7-s − 0.353i·8-s − 0.978·10-s + (0.540 + 0.841i)11-s − 0.179·13-s + (−0.267 − 0.654i)14-s + 0.250·16-s − 0.907·17-s + 0.413·19-s − 0.692i·20-s + (−0.595 + 0.381i)22-s − 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6507063962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6507063962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.44 - i)T \) |
| 11 | \( 1 + (-1.79 - 2.79i)T \) |
| good | 5 | \( 1 - 3.09iT - 5T^{2} \) |
| 13 | \( 1 + 0.646T + 13T^{2} \) |
| 17 | \( 1 + 3.74T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 1.58iT - 29T^{2} \) |
| 31 | \( 1 - 8.64iT - 31T^{2} \) |
| 37 | \( 1 - 3.58T + 37T^{2} \) |
| 41 | \( 1 + 9.93T + 41T^{2} \) |
| 43 | \( 1 + 7.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.93iT - 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 0.646iT - 59T^{2} \) |
| 61 | \( 1 + 1.93T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−10.13733176930392343447951681523, −9.260604341871924287198297700385, −8.507289241203888662154703784419, −7.26766563881385272740619811406, −6.87540261190206956336341811581, −6.29618453462404373361274986643, −5.30117404910659583566261169525, −4.08506533188069850925645417858, −3.21575577424505567524994307039, −2.14460513965739607571071282306,
0.27086454512546710710797861483, 1.37730285344933610776914106172, 2.80119288668105503570191762885, 3.92806769499810969378522327121, 4.52521928285749338906822457574, 5.65494422157657927797276995164, 6.39424113538556225222176769515, 7.63864350534590975596137392653, 8.534065785127994707698641621687, 9.177029496173396803987257738709
