| L(s) = 1 | + 6.13·3-s − 14.8·5-s − 19.8·7-s + 10.6·9-s + 55.2·11-s + 1.83·13-s − 91.1·15-s − 130.·17-s + 17.3·19-s − 121.·21-s − 23·23-s + 95.5·25-s − 100.·27-s − 77.6·29-s + 206.·31-s + 338.·33-s + 294.·35-s + 251.·37-s + 11.2·39-s − 157.·41-s + 336.·43-s − 157.·45-s + 488.·47-s + 51.2·49-s − 798.·51-s − 562.·53-s − 820.·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s − 1.32·5-s − 1.07·7-s + 0.393·9-s + 1.51·11-s + 0.0391·13-s − 1.56·15-s − 1.85·17-s + 0.208·19-s − 1.26·21-s − 0.208·23-s + 0.764·25-s − 0.716·27-s − 0.497·29-s + 1.19·31-s + 1.78·33-s + 1.42·35-s + 1.11·37-s + 0.0462·39-s − 0.598·41-s + 1.19·43-s − 0.522·45-s + 1.51·47-s + 0.149·49-s − 2.19·51-s − 1.45·53-s − 2.01·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.820150608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820150608\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 6.13T + 27T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 + 19.8T + 343T^{2} \) |
| 11 | \( 1 - 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.83T + 2.19e3T^{2} \) |
| 17 | \( 1 + 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 17.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 77.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 336.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 562.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 125.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 988.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.18e3T + 9.12e5T^{2} \) |
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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
−9.254424924620961385307549613914, −8.379759358302357595586987145074, −7.67392751654585182468758675870, −6.78728281928981502294331232085, −6.19628626488318017494447137400, −4.44176828928177770709578621535, −3.89916772054568449117780305025, −3.20224447976282832124567360278, −2.20519919010694955868953873730, −0.60354547107632897232473418140,
0.60354547107632897232473418140, 2.20519919010694955868953873730, 3.20224447976282832124567360278, 3.89916772054568449117780305025, 4.44176828928177770709578621535, 6.19628626488318017494447137400, 6.78728281928981502294331232085, 7.67392751654585182468758675870, 8.379759358302357595586987145074, 9.254424924620961385307549613914
