| L(s) = 1 | + 0.0889·2-s + 0.505·3-s − 1.99·4-s + 2.21·5-s + 0.0449·6-s − 0.354·8-s − 2.74·9-s + 0.196·10-s − 1.73·11-s − 1.00·12-s + 4.79·13-s + 1.11·15-s + 3.95·16-s + 3.38·17-s − 0.244·18-s − 6.16·19-s − 4.41·20-s − 0.154·22-s + 3.35·23-s − 0.179·24-s − 0.0908·25-s + 0.426·26-s − 2.90·27-s + 4.00·29-s + 0.0995·30-s − 6.35·31-s + 1.06·32-s + ⋯ |
| L(s) = 1 | + 0.0628·2-s + 0.291·3-s − 0.996·4-s + 0.990·5-s + 0.0183·6-s − 0.125·8-s − 0.914·9-s + 0.0622·10-s − 0.524·11-s − 0.290·12-s + 1.33·13-s + 0.289·15-s + 0.988·16-s + 0.820·17-s − 0.0575·18-s − 1.41·19-s − 0.986·20-s − 0.0329·22-s + 0.699·23-s − 0.0366·24-s − 0.0181·25-s + 0.0836·26-s − 0.558·27-s + 0.744·29-s + 0.0181·30-s − 1.14·31-s + 0.187·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909530844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909530844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0889T + 2T^{2} \) |
| 3 | \( 1 - 0.505T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 4.00T + 29T^{2} \) |
| 31 | \( 1 + 6.35T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 - 9.89T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 - 2.79T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 9.56T + 89T^{2} \) |
| 97 | \( 1 - 7.88T + 97T^{2} \) |
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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
−8.188904616829653603984040474794, −7.58268602518575953400184342955, −6.25893645141263433447733596515, −5.93006502191433807945147819491, −5.30686051937702728605623421501, −4.43541902755365100741306599021, −3.58848236543312762486795716956, −2.84821112890216892927173756001, −1.86481219931619800615271860591, −0.71309782594025108085957337319,
0.71309782594025108085957337319, 1.86481219931619800615271860591, 2.84821112890216892927173756001, 3.58848236543312762486795716956, 4.43541902755365100741306599021, 5.30686051937702728605623421501, 5.93006502191433807945147819491, 6.25893645141263433447733596515, 7.58268602518575953400184342955, 8.188904616829653603984040474794
