| L(s) = 1 | + 3-s − 3.50·5-s − 3.14·7-s + 9-s + 5.14·11-s + 3.14·13-s − 3.50·15-s + 3.08·17-s − 1.86·19-s − 3.14·21-s + 3.63·23-s + 7.28·25-s + 27-s + 3.50·29-s + 0.726·31-s + 5.14·33-s + 11.0·35-s − 37-s + 3.14·39-s + 2·41-s + 5.55·43-s − 3.50·45-s + 11.7·47-s + 2.86·49-s + 3.08·51-s + 1.68·53-s − 18.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.56·5-s − 1.18·7-s + 0.333·9-s + 1.55·11-s + 0.871·13-s − 0.904·15-s + 0.749·17-s − 0.428·19-s − 0.685·21-s + 0.758·23-s + 1.45·25-s + 0.192·27-s + 0.650·29-s + 0.130·31-s + 0.894·33-s + 1.86·35-s − 0.164·37-s + 0.503·39-s + 0.312·41-s + 0.847·43-s − 0.522·45-s + 1.71·47-s + 0.409·49-s + 0.432·51-s + 0.231·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.426995932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426995932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 - 3.08T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 0.726T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 7.73T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−10.06231107582857723563084528294, −9.005265989722145881406549123805, −8.649289350370241724490948080258, −7.53557902824715732623056730331, −6.87223543778920493057856843566, −6.00368065105927930086907282004, −4.30156304877550679324289641331, −3.74684073735669892405138841546, −2.99773605905308709412462327131, −0.969358935085209964984366932343,
0.969358935085209964984366932343, 2.99773605905308709412462327131, 3.74684073735669892405138841546, 4.30156304877550679324289641331, 6.00368065105927930086907282004, 6.87223543778920493057856843566, 7.53557902824715732623056730331, 8.649289350370241724490948080258, 9.005265989722145881406549123805, 10.06231107582857723563084528294
