| L(s) = 1 | − 2.78·3-s + 2.23i·5-s + 9.63i·7-s − 1.23·9-s + 8.11i·11-s − 3.04·13-s − 6.22i·15-s − 28.9i·17-s − 30.9i·19-s − 26.8i·21-s + (20.0 + 11.3i)23-s − 5.00·25-s + 28.5·27-s + 31.5·29-s + 51.7·31-s + ⋯ |
| L(s) = 1 | − 0.928·3-s + 0.447i·5-s + 1.37i·7-s − 0.137·9-s + 0.737i·11-s − 0.234·13-s − 0.415i·15-s − 1.70i·17-s − 1.62i·19-s − 1.27i·21-s + (0.870 + 0.492i)23-s − 0.200·25-s + 1.05·27-s + 1.08·29-s + 1.66·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.230179838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230179838\) |
| \(L(2)\) |
|
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 | \( 1 + (-20.0 - 11.3i)T \) |
| good | 3 | \( 1 + 2.78T + 9T^{2} \) |
| 7 | \( 1 - 9.63iT - 49T^{2} \) |
| 11 | \( 1 - 8.11iT - 121T^{2} \) |
| 13 | \( 1 + 3.04T + 169T^{2} \) |
| 17 | \( 1 + 28.9iT - 289T^{2} \) |
| 19 | \( 1 + 30.9iT - 361T^{2} \) |
| 29 | \( 1 - 31.5T + 841T^{2} \) |
| 31 | \( 1 - 51.7T + 961T^{2} \) |
| 37 | \( 1 + 15.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 29.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 35.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 13.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 132.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 61.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 90.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 5.60iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 72.1iT - 9.40e3T^{2} \) |
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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
−9.219536522108411353081092775162, −8.592022579877505580384885673908, −7.37404999192183027199612151383, −6.74804378827010565822442034438, −6.00250860966144719079084115427, −4.99061516380328534709878925269, −4.79998530476648205225176482601, −2.85149161998511983959638648924, −2.55497702244190501961451283331, −0.74433466568758264338384304337,
0.58409132012013826253248080912, 1.40011176850331215883638723219, 3.12621051572403539614766654224, 4.12476222157209990123612018660, 4.78707642190507313246679695663, 5.95263203523153359413439660463, 6.25221485681736742558717253994, 7.31888594823924223608263202371, 8.244435065610000716927071374348, 8.695234942687703267997240656240
