| L(s) = 1 | − 2.23i·3-s + 0.538i·5-s − 0.0679i·7-s − 2.00·9-s − 1.24i·11-s + 4.00·13-s + 1.20·15-s + (−2.13 + 3.52i)17-s − 3.80·19-s − 0.152·21-s + 7.87i·23-s + 4.71·25-s − 2.22i·27-s − 9.03i·29-s − 7.50i·31-s + ⋯ |
| L(s) = 1 | − 1.29i·3-s + 0.240i·5-s − 0.0256i·7-s − 0.668·9-s − 0.376i·11-s + 1.11·13-s + 0.311·15-s + (−0.518 + 0.854i)17-s − 0.872·19-s − 0.0331·21-s + 1.64i·23-s + 0.942·25-s − 0.428i·27-s − 1.67i·29-s − 1.34i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726171314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726171314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.13 - 3.52i)T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 5 | \( 1 - 0.538iT - 5T^{2} \) |
| 7 | \( 1 + 0.0679iT - 7T^{2} \) |
| 11 | \( 1 + 1.24iT - 11T^{2} \) |
| 13 | \( 1 - 4.00T + 13T^{2} \) |
| 19 | \( 1 + 3.80T + 19T^{2} \) |
| 23 | \( 1 - 7.87iT - 23T^{2} \) |
| 29 | \( 1 + 9.03iT - 29T^{2} \) |
| 31 | \( 1 + 7.50iT - 31T^{2} \) |
| 37 | \( 1 - 7.34iT - 37T^{2} \) |
| 41 | \( 1 + 11.9iT - 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 0.438T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 61 | \( 1 + 15.0iT - 61T^{2} \) |
| 67 | \( 1 + 4.19T + 67T^{2} \) |
| 71 | \( 1 - 7.28iT - 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.00iT - 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 6.93iT - 97T^{2} \) |
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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
−8.102936224828371337645770680312, −7.51373321161309306761019471137, −6.61496585190994446863839949007, −6.22608917068963903203217007667, −5.54127037695941852171883645769, −4.22490811182417164152712296868, −3.54249201749072308814634681411, −2.34702679870357860919787169903, −1.65335656354154905476761053145, −0.55136532014088175135313733865,
1.13405426583007718194400181505, 2.53571035985344540064188111079, 3.41124932740647355607150881096, 4.30540795972904918224346832029, 4.75539458196165031846085005928, 5.48309873737051217265964317231, 6.56145100513404740842823400404, 7.04305762996211683339708740475, 8.307471877457614081781778625524, 8.956582707868272725089703062364
