| L(s) = 1 | − 2·2-s − 2.09·3-s + 4·4-s − 4.70·5-s + 4.19·6-s + 7·7-s − 8·8-s − 22.5·9-s + 9.41·10-s − 31.8·11-s − 8.39·12-s − 14·14-s + 9.88·15-s + 16·16-s − 33.6·17-s + 45.1·18-s − 83.2·19-s − 18.8·20-s − 14.6·21-s + 63.7·22-s − 11.7·23-s + 16.7·24-s − 102.·25-s + 104.·27-s + 28·28-s − 275.·29-s − 19.7·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.403·3-s + 0.5·4-s − 0.421·5-s + 0.285·6-s + 0.377·7-s − 0.353·8-s − 0.836·9-s + 0.297·10-s − 0.874·11-s − 0.201·12-s − 0.267·14-s + 0.170·15-s + 0.250·16-s − 0.479·17-s + 0.591·18-s − 1.00·19-s − 0.210·20-s − 0.152·21-s + 0.618·22-s − 0.106·23-s + 0.142·24-s − 0.822·25-s + 0.741·27-s + 0.188·28-s − 1.76·29-s − 0.120·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1194689466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1194689466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 - 7T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.09T + 27T^{2} \) |
| 5 | \( 1 + 4.70T + 125T^{2} \) |
| 11 | \( 1 + 31.8T + 1.33e3T^{2} \) |
| 17 | \( 1 + 33.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 275.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 112.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 487.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 359.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 481.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 294.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 935.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 961.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 377.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 231.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 597.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.57e3T + 9.12e5T^{2} \) |
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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
−8.573219633548034396306846974985, −7.893305863039561420165538220770, −7.34750687393297453799215882308, −6.25052783781832585711412337927, −5.66347128757330277721586707197, −4.74052315973355834223341363678, −3.70102272290074327397773725387, −2.59779313657869640471284572270, −1.72333969997484305075650515690, −0.16682198511698577848817089620,
0.16682198511698577848817089620, 1.72333969997484305075650515690, 2.59779313657869640471284572270, 3.70102272290074327397773725387, 4.74052315973355834223341363678, 5.66347128757330277721586707197, 6.25052783781832585711412337927, 7.34750687393297453799215882308, 7.893305863039561420165538220770, 8.573219633548034396306846974985
