| L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−1.03 − 0.303i)5-s + (0.415 − 0.909i)6-s + (0.654 − 0.755i)7-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.706 + 0.815i)10-s + (2.22 − 1.43i)11-s + (−0.841 + 0.540i)12-s + (−3.65 − 4.21i)13-s + (−0.959 + 0.281i)14-s + (0.153 − 1.06i)15-s + (−0.654 + 0.755i)16-s + (−2.14 + 4.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (−0.462 − 0.135i)5-s + (0.169 − 0.371i)6-s + (0.247 − 0.285i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.223 + 0.257i)10-s + (0.672 − 0.431i)11-s + (−0.242 + 0.156i)12-s + (−1.01 − 1.17i)13-s + (−0.256 + 0.0752i)14-s + (0.0396 − 0.275i)15-s + (−0.163 + 0.188i)16-s + (−0.519 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.198760 - 0.453795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.198760 - 0.453795i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.02 - 4.68i)T \) |
| good | 5 | \( 1 + (1.03 + 0.303i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.22 + 1.43i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.65 + 4.21i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.14 - 4.68i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.750 + 1.64i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.478 - 1.04i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.22 + 8.49i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.690 - 0.202i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (8.20 + 2.40i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.658 + 4.57i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 + (-5.22 + 6.02i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (6.21 + 7.16i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.849 + 5.90i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-6.59 - 4.23i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (1.37 + 0.881i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.723 - 1.58i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (8.73 + 10.0i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.98 + 1.46i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.56 + 10.8i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (0.398 + 0.117i)T + (81.6 + 52.4i)T^{2} \) |
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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
−9.877997590113768638596182808121, −8.892089897465685452737561985497, −8.157768068942312293326389595103, −7.52626664309628627645250096335, −6.34179532647985868081433794029, −5.22416074969207375434341100504, −4.10714922077331344697667140346, −3.39264227606392518981810121118, −2.01614623887819740993470975612, −0.27012654076796852436137027927,
1.58809715552557938381873370527, 2.66460018433838993437391588780, 4.25460245087642659178475489342, 5.14813814717815660086817742348, 6.51699138777473441339211676210, 6.95669252141218567826733194595, 7.73587210646046621357368343862, 8.665679417689991905716005583391, 9.308442750103735071555636307971, 10.12233912403017293943145304267
