| L(s) = 1 | − 0.662i·3-s + (−1.31 + 1.81i)5-s + (1.19 − 2.35i)7-s + 2.56·9-s − 3.09i·11-s + 4.66·13-s + (1.19 + 0.868i)15-s + 2.04·17-s − 5.60·19-s + (−1.56 − 0.794i)21-s + 1.87·23-s + (−1.56 − 4.74i)25-s − 3.68i·27-s + 3.56·29-s − 8.74·31-s + ⋯ |
| L(s) = 1 | − 0.382i·3-s + (−0.586 + 0.810i)5-s + (0.453 − 0.891i)7-s + 0.853·9-s − 0.932i·11-s + 1.29·13-s + (0.309 + 0.224i)15-s + 0.496·17-s − 1.28·19-s + (−0.340 − 0.173i)21-s + 0.390·23-s + (−0.312 − 0.949i)25-s − 0.708i·27-s + 0.661·29-s − 1.57·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.796514221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796514221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.31 - 1.81i)T \) |
| 7 | \( 1 + (-1.19 + 2.35i)T \) |
| good | 3 | \( 1 + 0.662iT - 3T^{2} \) |
| 11 | \( 1 + 3.09iT - 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 - 2.04T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 3.56T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 3.70iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 + 0.290iT - 47T^{2} \) |
| 53 | \( 1 + 9.49iT - 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 - 2.39T + 67T^{2} \) |
| 71 | \( 1 + 9.65iT - 71T^{2} \) |
| 73 | \( 1 - 4.09T + 73T^{2} \) |
| 79 | \( 1 + 1.35iT - 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 6.14T + 97T^{2} \) |
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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
−8.573951346492968509588232999992, −8.109715176299651513382488124585, −7.33517327030460230050447672178, −6.63719150293184827968576226748, −6.05661145586182404819176539410, −4.71829295681076076146121753833, −3.85782515237680222571211678178, −3.29467328630885698546588446664, −1.83116542936271490287753311157, −0.72168060953063614358245303569,
1.23003704874057708375918364278, 2.21042182350279605361157573626, 3.76235925779948470215227588773, 4.25265483684572048667983695115, 5.13170903940784373729820802725, 5.83092446805149845231182605265, 6.97372258824980263550616213078, 7.69524002617489323633700377421, 8.654809604816702944449175792011, 8.936957909864171336273498863654
