| L(s) = 1 | + 3.07i·3-s + (−1.47 − 1.47i)7-s − 6.45·9-s + (1.20 + 1.20i)11-s − 5.63·13-s + (−4.22 − 4.22i)17-s + (3.11 + 3.11i)19-s + (4.54 − 4.54i)21-s + (1.08 − 1.08i)23-s − 10.6i·27-s + (5.32 − 5.32i)29-s − 4.67i·31-s + (−3.71 + 3.71i)33-s + 1.51·37-s − 17.3i·39-s + ⋯ |
| L(s) = 1 | + 1.77i·3-s + (−0.558 − 0.558i)7-s − 2.15·9-s + (0.363 + 0.363i)11-s − 1.56·13-s + (−1.02 − 1.02i)17-s + (0.715 + 0.715i)19-s + (0.991 − 0.991i)21-s + (0.225 − 0.225i)23-s − 2.04i·27-s + (0.988 − 0.988i)29-s − 0.839i·31-s + (−0.646 + 0.646i)33-s + 0.248·37-s − 2.77i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3562217185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3562217185\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.07iT - 3T^{2} \) |
| 7 | \( 1 + (1.47 + 1.47i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.20 - 1.20i)T + 11iT^{2} \) |
| 13 | \( 1 + 5.63T + 13T^{2} \) |
| 17 | \( 1 + (4.22 + 4.22i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.11 - 3.11i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.08 + 1.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (-5.32 + 5.32i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.67iT - 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 - 3.19iT - 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 + (0.827 - 0.827i)T - 47iT^{2} \) |
| 53 | \( 1 + 8.17iT - 53T^{2} \) |
| 59 | \( 1 + (7.78 - 7.78i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.03 + 3.03i)T + 61iT^{2} \) |
| 67 | \( 1 + 2.93T + 67T^{2} \) |
| 71 | \( 1 - 0.180T + 71T^{2} \) |
| 73 | \( 1 + (2.19 + 2.19i)T + 73iT^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.33iT - 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + (-6.04 - 6.04i)T + 97iT^{2} \) |
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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
−9.665357455089335591538342491951, −8.798466170091192344530545742074, −7.70274312900655029924685448301, −6.83783576234691434700019870676, −5.80773158550171582074302436718, −4.69062353492748511347532985893, −4.48444790496686946679194538411, −3.36373941914839579352428341168, −2.51635366610193654896655101820, −0.13682325170467644433210323493,
1.32867476037306810841485263115, 2.43841533302217117486761629281, 3.11595426282638111781239595303, 4.76561391753479509092165465003, 5.74519028390834527064815467543, 6.56201969713712441572345781160, 7.03159696173865203223207411370, 7.80369381573592310921680631730, 8.737806976312335132346558062614, 9.203346635394804818086722176635
