L(s) = 1 | − 3·3-s + 0.612·5-s + 22.7·7-s + 9·9-s + 60.2·11-s + 52.9·13-s − 1.83·15-s + 47.1·17-s + 29.1·19-s − 68.2·21-s − 109.·23-s − 124.·25-s − 27·27-s − 10.4·29-s − 220.·31-s − 180.·33-s + 13.9·35-s + 408.·37-s − 158.·39-s + 360.·41-s − 236.·43-s + 5.51·45-s + 129.·47-s + 174.·49-s − 141.·51-s + 117.·53-s + 36.9·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.0547·5-s + 1.22·7-s + 0.333·9-s + 1.65·11-s + 1.12·13-s − 0.0316·15-s + 0.672·17-s + 0.352·19-s − 0.709·21-s − 0.992·23-s − 0.996·25-s − 0.192·27-s − 0.0667·29-s − 1.27·31-s − 0.953·33-s + 0.0672·35-s + 1.81·37-s − 0.651·39-s + 1.37·41-s − 0.838·43-s + 0.0182·45-s + 0.400·47-s + 0.508·49-s − 0.388·51-s + 0.305·53-s + 0.0905·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.417505464\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.417505464\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
good | 5 | \( 1 - 0.612T + 125T^{2} \) |
| 7 | \( 1 - 22.7T + 343T^{2} \) |
| 11 | \( 1 - 60.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 40.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 614.T + 9.12e5T^{2} \) |
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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
−9.915433652230749060032508076354, −9.122725069709323141008836622903, −8.159578213859178123920310894053, −7.38704325984343857196157800358, −6.19358130390038909314243512682, −5.66071443774082057621598224566, −4.38332206022596767761046015754, −3.71734739814407388335297811575, −1.81637270437719407369744240494, −1.00768605250035216649018847832,
1.00768605250035216649018847832, 1.81637270437719407369744240494, 3.71734739814407388335297811575, 4.38332206022596767761046015754, 5.66071443774082057621598224566, 6.19358130390038909314243512682, 7.38704325984343857196157800358, 8.159578213859178123920310894053, 9.122725069709323141008836622903, 9.915433652230749060032508076354