| L(s) = 1 | + 1.53·2-s + 2.87·3-s + 0.347·4-s + 2.34·5-s + 4.41·6-s − 2.53·8-s + 5.29·9-s + 3.59·10-s + 0.999·12-s + 0.184·13-s + 6.75·15-s − 4.57·16-s + 3.92·17-s + 8.10·18-s − 0.773·19-s + 0.815·20-s + 8.35·23-s − 7.29·24-s + 0.509·25-s + 0.283·26-s + 6.59·27-s + 8.17·29-s + 10.3·30-s + 2.65·31-s − 1.94·32-s + 6.00·34-s + 1.83·36-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 1.66·3-s + 0.173·4-s + 1.04·5-s + 1.80·6-s − 0.895·8-s + 1.76·9-s + 1.13·10-s + 0.288·12-s + 0.0512·13-s + 1.74·15-s − 1.14·16-s + 0.951·17-s + 1.91·18-s − 0.177·19-s + 0.182·20-s + 1.74·23-s − 1.48·24-s + 0.101·25-s + 0.0555·26-s + 1.26·27-s + 1.51·29-s + 1.89·30-s + 0.476·31-s − 0.343·32-s + 1.03·34-s + 0.306·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.581317577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.581317577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 13 | \( 1 - 0.184T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 19 | \( 1 + 0.773T + 19T^{2} \) |
| 23 | \( 1 - 8.35T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 2.65T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 + 0.426T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 - 6.55T + 53T^{2} \) |
| 59 | \( 1 - 0.204T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 3.75T + 67T^{2} \) |
| 71 | \( 1 + 9.96T + 71T^{2} \) |
| 73 | \( 1 + 0.120T + 73T^{2} \) |
| 79 | \( 1 + 0.327T + 79T^{2} \) |
| 83 | \( 1 + 3.35T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.274636553197369429445910950182, −7.31861840330761468627835599752, −6.60864333738901649989456481813, −5.84343487448261527971310833784, −5.01953325621758552934962387369, −4.44750952169204532347612821671, −3.35388908636984612911433313909, −3.07089669641036519316855099724, −2.26383317957955641376654492355, −1.27283280644566572789217725769,
1.27283280644566572789217725769, 2.26383317957955641376654492355, 3.07089669641036519316855099724, 3.35388908636984612911433313909, 4.44750952169204532347612821671, 5.01953325621758552934962387369, 5.84343487448261527971310833784, 6.60864333738901649989456481813, 7.31861840330761468627835599752, 8.274636553197369429445910950182
