| L(s) = 1 | + 2.82·3-s + 5-s + 2·7-s + 5.00·9-s + 11-s + 6.82·13-s + 2.82·15-s + 1.17·17-s + 5.65·21-s − 2.82·23-s + 25-s + 5.65·27-s − 7.65·29-s + 2.82·33-s + 2·35-s − 3.65·37-s + 19.3·39-s + 6·41-s − 6·43-s + 5.00·45-s + 2.82·47-s − 3·49-s + 3.31·51-s − 0.343·53-s + 55-s − 9.65·59-s − 13.3·61-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.447·5-s + 0.755·7-s + 1.66·9-s + 0.301·11-s + 1.89·13-s + 0.730·15-s + 0.284·17-s + 1.23·21-s − 0.589·23-s + 0.200·25-s + 1.08·27-s − 1.42·29-s + 0.492·33-s + 0.338·35-s − 0.601·37-s + 3.09·39-s + 0.937·41-s − 0.914·43-s + 0.745·45-s + 0.412·47-s − 0.428·49-s + 0.464·51-s − 0.0471·53-s + 0.134·55-s − 1.25·59-s − 1.70·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.593902067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.593902067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 0.343T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.48T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.610291951671636980696741646138, −7.982634831305173861890638016802, −7.40772205496255758928675159377, −6.35078654501034359110418518726, −5.64662325074703575195718143094, −4.47703088115037419666106377574, −3.72289347911797440731423747855, −3.10082749812985903902254986921, −1.90958965765353704993650312158, −1.42451730871385649144974514127,
1.42451730871385649144974514127, 1.90958965765353704993650312158, 3.10082749812985903902254986921, 3.72289347911797440731423747855, 4.47703088115037419666106377574, 5.64662325074703575195718143094, 6.35078654501034359110418518726, 7.40772205496255758928675159377, 7.982634831305173861890638016802, 8.610291951671636980696741646138
