| L(s) = 1 | + i·2-s − 4-s − 1.18i·5-s + (−1.74 + 1.99i)7-s − i·8-s + 1.18·10-s + (2.25 − 2.43i)11-s − 5.98·13-s + (−1.99 − 1.74i)14-s + 16-s + 4.29·17-s + 0.203·19-s + 1.18i·20-s + (2.43 + 2.25i)22-s + 3.11·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.530i·5-s + (−0.658 + 0.752i)7-s − 0.353i·8-s + 0.374·10-s + (0.678 − 0.734i)11-s − 1.65·13-s + (−0.532 − 0.465i)14-s + 0.250·16-s + 1.04·17-s + 0.0467·19-s + 0.265i·20-s + (0.519 + 0.479i)22-s + 0.648·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342394745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342394745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.74 - 1.99i)T \) |
| 11 | \( 1 + (-2.25 + 2.43i)T \) |
| good | 5 | \( 1 + 1.18iT - 5T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 - 0.203T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 + 2.50iT - 29T^{2} \) |
| 31 | \( 1 + 2.79iT - 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 2.05iT - 47T^{2} \) |
| 53 | \( 1 + 1.51T + 53T^{2} \) |
| 59 | \( 1 + 7.96iT - 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 - 9.85iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 8.87iT - 89T^{2} \) |
| 97 | \( 1 + 9.46iT - 97T^{2} \) |
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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
−9.514273573954304940430141528774, −8.785106946700439031696990005573, −7.981947922625733890799201307702, −7.12572849913150823629923350271, −6.26383156278197448682750462787, −5.48003014757622995947781611906, −4.78060137170824253532304702118, −3.58340902669046979669241260821, −2.51323871052687212066737332334, −0.69460519211885189528519600014,
1.04448558181493174914364809438, 2.52701102182377254967393539240, 3.30409607505141278872047827888, 4.33700538294574724969752316918, 5.13416354358247769810151909521, 6.43942066540512543558108968833, 7.18984158767925107411780493749, 7.77337986900732340440720627713, 9.164478491543966947137777361319, 9.709944050480910142211400940557
