| L(s) = 1 | + (1.30 − 1.14i)3-s + 5-s + 3.38i·7-s + (0.381 − 2.97i)9-s + 1.64·11-s − 5.60i·13-s + (1.30 − 1.14i)15-s + 6.51i·17-s − 0.200·19-s + (3.87 + 4.39i)21-s − 6.34i·23-s + 25-s + (−2.90 − 4.30i)27-s − 6.66i·29-s + 4.74i·31-s + ⋯ |
| L(s) = 1 | + (0.750 − 0.660i)3-s + 0.447·5-s + 1.27i·7-s + (0.127 − 0.991i)9-s + 0.495·11-s − 1.55i·13-s + (0.335 − 0.295i)15-s + 1.57i·17-s − 0.0460·19-s + (0.844 + 0.959i)21-s − 1.32i·23-s + 0.200·25-s + (−0.559 − 0.828i)27-s − 1.23i·29-s + 0.853i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913894501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913894501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (0.760 + 8.14i)T \) |
| good | 7 | \( 1 - 3.38iT - 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + 5.60iT - 13T^{2} \) |
| 17 | \( 1 - 6.51iT - 17T^{2} \) |
| 19 | \( 1 + 0.200T + 19T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 + 6.66iT - 29T^{2} \) |
| 31 | \( 1 - 4.74iT - 31T^{2} \) |
| 37 | \( 1 - 8.17T + 37T^{2} \) |
| 41 | \( 1 - 7.39T + 41T^{2} \) |
| 43 | \( 1 - 4.85iT - 43T^{2} \) |
| 47 | \( 1 + 6.48iT - 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 2.26iT - 59T^{2} \) |
| 61 | \( 1 + 3.37iT - 61T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 9.54iT - 79T^{2} \) |
| 83 | \( 1 - 3.35iT - 83T^{2} \) |
| 89 | \( 1 - 8.56iT - 89T^{2} \) |
| 97 | \( 1 + 0.600iT - 97T^{2} \) |
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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
−8.220533818361211105302240947638, −8.011089786648303108340139778706, −6.79859692354631145231383401212, −6.02158181254931890967413437743, −5.78586636737989142676898458345, −4.52609840667138641384415910109, −3.50120404624317860301244432529, −2.63525002729482298191537122563, −2.06920524210081810125532495648, −0.868666529256830340683438602528,
1.11928437492380367485428898154, 2.16312113340743484313319753606, 3.16301370968735848905739626803, 4.07088280449979109035811369396, 4.48467968474795233819887868095, 5.39466070770344786845156472365, 6.47000419840052596586173669288, 7.35855055794014059847747797927, 7.52620868482778213029629793830, 8.802964189762126982762652706437
