| L(s) = 1 | + 3-s + 2.72i·5-s − 3.31i·7-s + 9-s − 0.656i·11-s + 2.72i·15-s + 1.16·17-s + 6.77i·19-s − 3.31i·21-s − 6.90·23-s − 2.41·25-s + 27-s − 5.82·29-s − 0.969i·31-s − 0.656i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.21i·5-s − 1.25i·7-s + 0.333·9-s − 0.197i·11-s + 0.702i·15-s + 0.283·17-s + 1.55i·19-s − 0.723i·21-s − 1.43·23-s − 0.482·25-s + 0.192·27-s − 1.08·29-s − 0.174i·31-s − 0.114i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0304 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0304 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868949130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868949130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2.72iT - 5T^{2} \) |
| 7 | \( 1 + 3.31iT - 7T^{2} \) |
| 11 | \( 1 + 0.656iT - 11T^{2} \) |
| 17 | \( 1 - 1.16T + 17T^{2} \) |
| 19 | \( 1 - 6.77iT - 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 0.969iT - 31T^{2} \) |
| 37 | \( 1 - 9.93iT - 37T^{2} \) |
| 41 | \( 1 - 10.5iT - 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 - 0.841T + 53T^{2} \) |
| 59 | \( 1 + 0.128iT - 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 15.9iT - 67T^{2} \) |
| 71 | \( 1 - 5.32iT - 71T^{2} \) |
| 73 | \( 1 - 3.75iT - 73T^{2} \) |
| 79 | \( 1 - 2.17T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 4.09iT - 89T^{2} \) |
| 97 | \( 1 - 16.5iT - 97T^{2} \) |
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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
−8.337058295325985067454991362184, −7.79426662017725141187030961106, −7.34865979080855241933101065337, −6.43799784891908845792945126999, −5.96537416778393839611278372028, −4.64403254576940455664897947058, −3.74411118563819736610574531007, −3.39291659800115939842678142974, −2.31939874404741622707551680611, −1.26053991831433699234992017654,
0.49456822188778008966852530944, 1.93944147544407252290693041186, 2.45509081964461009773657973116, 3.70903897850426779505256384486, 4.44032627063479792887088818579, 5.45073168847856937579746804405, 5.65827577103269312773067670067, 6.94102322223682007774305815812, 7.62327025594594364916819990360, 8.625954389449413275057165776246
