L(s) = 1 | − 4.08·2-s + 3·3-s + 8.72·4-s − 14.1·5-s − 12.2·6-s − 18.6·7-s − 2.97·8-s + 9·9-s + 57.9·10-s + 11·11-s + 26.1·12-s + 63.5·13-s + 76.3·14-s − 42.5·15-s − 57.6·16-s + 65.4·17-s − 36.8·18-s − 27.2·19-s − 123.·20-s − 56.0·21-s − 44.9·22-s + 43.8·23-s − 8.91·24-s + 75.7·25-s − 259.·26-s + 27·27-s − 162.·28-s + ⋯ |
L(s) = 1 | − 1.44·2-s + 0.577·3-s + 1.09·4-s − 1.26·5-s − 0.834·6-s − 1.00·7-s − 0.131·8-s + 0.333·9-s + 1.83·10-s + 0.301·11-s + 0.629·12-s + 1.35·13-s + 1.45·14-s − 0.731·15-s − 0.900·16-s + 0.934·17-s − 0.481·18-s − 0.329·19-s − 1.38·20-s − 0.581·21-s − 0.435·22-s + 0.397·23-s − 0.0758·24-s + 0.605·25-s − 1.95·26-s + 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7819217847\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7819217847\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 4.08T + 8T^{2} \) |
| 5 | \( 1 + 14.1T + 125T^{2} \) |
| 7 | \( 1 + 18.6T + 343T^{2} \) |
| 13 | \( 1 - 63.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 43.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 341.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 6.98T + 5.06e4T^{2} \) |
| 41 | \( 1 - 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 301.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 83.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 66.5T + 2.05e5T^{2} \) |
| 67 | \( 1 + 443.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 482.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 177.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 211.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 948.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 680.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−8.676639281255520651023333568544, −8.215285141423364791815632443745, −7.53660170836948272551638073883, −6.82800950451674832558820120384, −5.99845612738679705446100149250, −4.40596514424173068021005599016, −3.65184900808710959552527623359, −2.87635100828853936706817250250, −1.41215980518246048436209557348, −0.53699641831291727239177828288,
0.53699641831291727239177828288, 1.41215980518246048436209557348, 2.87635100828853936706817250250, 3.65184900808710959552527623359, 4.40596514424173068021005599016, 5.99845612738679705446100149250, 6.82800950451674832558820120384, 7.53660170836948272551638073883, 8.215285141423364791815632443745, 8.676639281255520651023333568544