| L(s) = 1 | + 3.86i·5-s − i·7-s − 5.27·11-s − 2·13-s − 1.03i·17-s − 5.46i·19-s + 0.378·23-s − 9.92·25-s + 6.31i·29-s − 9.46i·31-s + 3.86·35-s + 10.9·37-s − 5.93i·41-s + 4i·43-s + 8.48·47-s + ⋯ |
| L(s) = 1 | + 1.72i·5-s − 0.377i·7-s − 1.59·11-s − 0.554·13-s − 0.251i·17-s − 1.25i·19-s + 0.0790·23-s − 1.98·25-s + 1.17i·29-s − 1.69i·31-s + 0.653·35-s + 1.79·37-s − 0.926i·41-s + 0.609i·43-s + 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.096195471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096195471\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.86iT - 5T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 + 5.46iT - 19T^{2} \) |
| 23 | \( 1 - 0.378T + 23T^{2} \) |
| 29 | \( 1 - 6.31iT - 29T^{2} \) |
| 31 | \( 1 + 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.93iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 7.07iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 + 15.4iT - 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 3.86iT - 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.077619469314203923339736876908, −7.47185999343132650050564384683, −7.08889013399475568748332335721, −6.27621538954624375512192677478, −5.44759858082785900586111586822, −4.60528992304404352742537415589, −3.58102387657258054382917877225, −2.62756090221808372127290672174, −2.40182569891396509306676571593, −0.37708266652230527765641067874,
0.898709183509367517827594519479, 2.00040718461364045831048168383, 2.93046905773789591252870197995, 4.23072478768187659294898867332, 4.74433780983306276472310746011, 5.61801873088495844012043094267, 5.86975298723965075519239918789, 7.31277756320109082986696791735, 7.979973805879425871499235579970, 8.434263168648569862009463589286
