L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 4.03·5-s + 6·6-s + 17.2·7-s − 8·8-s + 9·9-s + 8.07·10-s − 41.8·11-s − 12·12-s + 0.844·13-s − 34.5·14-s + 12.1·15-s + 16·16-s − 15.4·17-s − 18·18-s − 16.1·20-s − 51.8·21-s + 83.7·22-s + 165.·23-s + 24·24-s − 108.·25-s − 1.68·26-s − 27·27-s + 69.0·28-s + 21.1·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.361·5-s + 0.408·6-s + 0.932·7-s − 0.353·8-s + 0.333·9-s + 0.255·10-s − 1.14·11-s − 0.288·12-s + 0.0180·13-s − 0.659·14-s + 0.208·15-s + 0.250·16-s − 0.220·17-s − 0.235·18-s − 0.180·20-s − 0.538·21-s + 0.812·22-s + 1.49·23-s + 0.204·24-s − 0.869·25-s − 0.0127·26-s − 0.192·27-s + 0.466·28-s + 0.135·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8913269599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8913269599\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 4.03T + 125T^{2} \) |
| 7 | \( 1 - 17.2T + 343T^{2} \) |
| 11 | \( 1 + 41.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.844T + 2.19e3T^{2} \) |
| 17 | \( 1 + 15.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 21.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 104.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 107.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 332.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 101.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 335.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 360.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 535.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 894.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 17.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 639.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−8.597023044424565991368174652441, −7.942893099241308531216830324782, −7.34150245031946590649725192819, −6.55147754136485006126303485149, −5.39833978737353349891519437403, −4.98683065930767183774346297006, −3.82213011622643716228004798695, −2.61771973559482962133145457059, −1.60500753538807006971793297552, −0.49856511756808975981397310790,
0.49856511756808975981397310790, 1.60500753538807006971793297552, 2.61771973559482962133145457059, 3.82213011622643716228004798695, 4.98683065930767183774346297006, 5.39833978737353349891519437403, 6.55147754136485006126303485149, 7.34150245031946590649725192819, 7.942893099241308531216830324782, 8.597023044424565991368174652441