| L(s) = 1 | + 2-s + 2.34·3-s + 4-s + 4.38·5-s + 2.34·6-s − 0.482·7-s + 8-s + 2.51·9-s + 4.38·10-s + 2.34·12-s − 5.73·13-s − 0.482·14-s + 10.2·15-s + 16-s + 4.18·17-s + 2.51·18-s + 19-s + 4.38·20-s − 1.13·21-s − 1.96·23-s + 2.34·24-s + 14.2·25-s − 5.73·26-s − 1.13·27-s − 0.482·28-s + 0.168·29-s + 10.2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.35·3-s + 0.5·4-s + 1.96·5-s + 0.958·6-s − 0.182·7-s + 0.353·8-s + 0.839·9-s + 1.38·10-s + 0.678·12-s − 1.58·13-s − 0.129·14-s + 2.65·15-s + 0.250·16-s + 1.01·17-s + 0.593·18-s + 0.229·19-s + 0.980·20-s − 0.247·21-s − 0.409·23-s + 0.479·24-s + 2.84·25-s − 1.12·26-s − 0.218·27-s − 0.0912·28-s + 0.0312·29-s + 1.87·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.952580704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.952580704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 5 | \( 1 - 4.38T + 5T^{2} \) |
| 7 | \( 1 + 0.482T + 7T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 4.18T + 17T^{2} \) |
| 23 | \( 1 + 1.96T + 23T^{2} \) |
| 29 | \( 1 - 0.168T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 3.59T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 4.21T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 - 0.314T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 - 2.20T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
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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
−8.188953215131255105954377059688, −7.68494081996392927860707321889, −6.72208762643286914001392118485, −6.13210659523476412886912245614, −5.26757836078792562704416603642, −4.75663219899512560125587348929, −3.49480851783641473123087468835, −2.73265180427454296010832654275, −2.30633759445980750002686321381, −1.43597856281681270903327205740,
1.43597856281681270903327205740, 2.30633759445980750002686321381, 2.73265180427454296010832654275, 3.49480851783641473123087468835, 4.75663219899512560125587348929, 5.26757836078792562704416603642, 6.13210659523476412886912245614, 6.72208762643286914001392118485, 7.68494081996392927860707321889, 8.188953215131255105954377059688
