| L(s) = 1 | + 0.869·3-s + 2.23i·5-s − 5.06i·7-s − 8.24·9-s + 7.85i·11-s − 10.3·13-s + 1.94i·15-s − 8.59i·17-s + 5.22i·19-s − 4.40i·21-s + (−15.1 + 17.2i)23-s − 5.00·25-s − 14.9·27-s + 53.1·29-s + 29.4·31-s + ⋯ |
| L(s) = 1 | + 0.289·3-s + 0.447i·5-s − 0.723i·7-s − 0.915·9-s + 0.714i·11-s − 0.797·13-s + 0.129i·15-s − 0.505i·17-s + 0.275i·19-s − 0.209i·21-s + (−0.659 + 0.751i)23-s − 0.200·25-s − 0.555·27-s + 1.83·29-s + 0.951·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.642739435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.642739435\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (15.1 - 17.2i)T \) |
| good | 3 | \( 1 - 0.869T + 9T^{2} \) |
| 7 | \( 1 + 5.06iT - 49T^{2} \) |
| 11 | \( 1 - 7.85iT - 121T^{2} \) |
| 13 | \( 1 + 10.3T + 169T^{2} \) |
| 17 | \( 1 + 8.59iT - 289T^{2} \) |
| 19 | \( 1 - 5.22iT - 361T^{2} \) |
| 29 | \( 1 - 53.1T + 841T^{2} \) |
| 31 | \( 1 - 29.4T + 961T^{2} \) |
| 37 | \( 1 + 45.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 48.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 65.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 9.26T + 2.20e3T^{2} \) |
| 53 | \( 1 - 51.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 52.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 115. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 5.12iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 37.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 96.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.481iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 37.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 116. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 54.7iT - 9.40e3T^{2} \) |
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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
−9.059930658493172030404870443517, −8.015780248622573890010038591652, −7.48836939186564912415255203176, −6.71547982954731642297602521212, −5.79688437376242247585428872028, −4.82137285514430321713027646437, −3.94305683392470856267601105109, −2.91977957140096327916683505017, −2.11645822423995391204082080784, −0.51552387984996519316937956981,
0.876199107423478259179567078471, 2.44149330646706794180733720904, 2.95158924294442457042503791262, 4.28565879468796470039293434145, 5.10393146743114957205000691938, 5.99295429901626161893142432177, 6.57372526305840548286080800763, 8.013722465245329459661256547938, 8.304660720015099899802433423951, 9.015046344712645813884492112075
