| L(s) = 1 | − 2.51i·3-s + (−1.92 + 1.13i)5-s − 4.64i·7-s − 3.32·9-s + 1.64·11-s + 1.91i·13-s + (2.85 + 4.84i)15-s − 0.969i·17-s − 4.91·19-s − 11.6·21-s − i·23-s + (2.42 − 4.37i)25-s + 0.825i·27-s − 7.48·29-s − 7.77·31-s + ⋯ |
| L(s) = 1 | − 1.45i·3-s + (−0.861 + 0.507i)5-s − 1.75i·7-s − 1.10·9-s + 0.497·11-s + 0.531i·13-s + (0.737 + 1.25i)15-s − 0.235i·17-s − 1.12·19-s − 2.54·21-s − 0.208i·23-s + (0.484 − 0.874i)25-s + 0.158i·27-s − 1.38·29-s − 1.39·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4968896719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4968896719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.92 - 1.13i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.51iT - 3T^{2} \) |
| 7 | \( 1 + 4.64iT - 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 1.91iT - 13T^{2} \) |
| 17 | \( 1 + 0.969iT - 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 29 | \( 1 + 7.48T + 29T^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 + 7.63iT - 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 3.77iT - 43T^{2} \) |
| 47 | \( 1 + 2.22iT - 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 - 12.0iT - 67T^{2} \) |
| 71 | \( 1 + 3.01T + 71T^{2} \) |
| 73 | \( 1 - 16.6iT - 73T^{2} \) |
| 79 | \( 1 + 0.163T + 79T^{2} \) |
| 83 | \( 1 - 5.66iT - 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 + 1.79iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.411523879427727337537170225133, −7.64869007376296826119311054774, −7.00718098005757999512078545475, −6.91520441828645483109253338823, −5.79011515598005953058453717793, −4.13712575539966498016506387436, −3.92063541066802323555310319936, −2.44365502649084994924220033570, −1.29547439711868513604474785249, −0.19017355555301537217612454725,
2.03886944756705237537917863312, 3.34266691492945401110820064807, 3.93267034635741324815670278759, 4.95674362587552535375056111075, 5.43738663000797774391253721950, 6.34918073758110925545117192255, 7.68781141756687540576361515545, 8.557674356238466238511845656502, 9.007527408416018541495190192978, 9.527112899346983569932053485264
