| L(s) = 1 | + 2.21i·3-s + 5.06i·7-s − 1.90·9-s − 2.55·11-s − 4.93i·13-s − 2.68i·17-s − 4.72·19-s − 11.2·21-s − 1.49i·23-s + 2.41i·27-s − 2.43·29-s + 3.19·31-s − 5.67i·33-s − 2.90i·37-s + 10.9·39-s + ⋯ |
| L(s) = 1 | + 1.27i·3-s + 1.91i·7-s − 0.636·9-s − 0.771·11-s − 1.36i·13-s − 0.651i·17-s − 1.08·19-s − 2.44·21-s − 0.311i·23-s + 0.465i·27-s − 0.451·29-s + 0.573·31-s − 0.987i·33-s − 0.478i·37-s + 1.75·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2805154092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2805154092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 2.21iT - 3T^{2} \) |
| 7 | \( 1 - 5.06iT - 7T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 + 4.93iT - 13T^{2} \) |
| 17 | \( 1 + 2.68iT - 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 1.49iT - 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 2.90iT - 37T^{2} \) |
| 43 | \( 1 + 7.62iT - 43T^{2} \) |
| 47 | \( 1 + 5.15iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.49T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 + 10.5iT - 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.28iT - 73T^{2} \) |
| 79 | \( 1 - 4.28T + 79T^{2} \) |
| 83 | \( 1 - 10.5iT - 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 3.37iT - 97T^{2} \) |
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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
−8.524121729601726445899243735340, −7.85451807921042455915256289369, −6.68833894505254560158593753280, −5.67630851728877711506390553157, −5.34551333896264484511280943542, −4.72171079604294707217546263522, −3.61991917364469964722336342811, −2.82043650841908994701810052938, −2.16691143668347396865640440920, −0.079723207578526011100499935300,
1.19226306949221549833061345847, 1.82520777247152123741553858143, 2.98717680689894999762254306896, 4.24901197689745335932155434781, 4.44885601411351677678900631740, 5.93183669271804193892566885154, 6.60340729685096424960021235362, 7.07825926948405559634864023052, 7.73841724511835007547761677733, 8.185522916081055541125251910652
