| L(s) = 1 | − 1.70·2-s − 5.10·4-s − 7·7-s + 22.2·8-s − 37.4·11-s − 29.0·13-s + 11.9·14-s + 2.89·16-s + 58.4·17-s − 54.5·19-s + 63.6·22-s + 161.·23-s + 49.3·26-s + 35.7·28-s − 137.·29-s + 154.·31-s − 183.·32-s − 99.4·34-s + 350.·37-s + 92.8·38-s − 353.·41-s + 518.·43-s + 190.·44-s − 275.·46-s − 542.·47-s + 49·49-s + 148.·52-s + ⋯ |
| L(s) = 1 | − 0.601·2-s − 0.638·4-s − 0.377·7-s + 0.985·8-s − 1.02·11-s − 0.619·13-s + 0.227·14-s + 0.0452·16-s + 0.833·17-s − 0.659·19-s + 0.616·22-s + 1.46·23-s + 0.372·26-s + 0.241·28-s − 0.880·29-s + 0.896·31-s − 1.01·32-s − 0.501·34-s + 1.55·37-s + 0.396·38-s − 1.34·41-s + 1.83·43-s + 0.654·44-s − 0.881·46-s − 1.68·47-s + 0.142·49-s + 0.394·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 11 | \( 1 + 37.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 29.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 137.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 350.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 518.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 542.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 305.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 14.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 171.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 377.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−8.761543260320092012523303247857, −7.84234479147763474799855470839, −7.41779908318893452241332078150, −6.25544222955419009800857274325, −5.21806612470637396149750558220, −4.62089319860350269156947609330, −3.44510499800316644381209374912, −2.42003280597919928770959819897, −1.00965334270187999610832995518, 0,
1.00965334270187999610832995518, 2.42003280597919928770959819897, 3.44510499800316644381209374912, 4.62089319860350269156947609330, 5.21806612470637396149750558220, 6.25544222955419009800857274325, 7.41779908318893452241332078150, 7.84234479147763474799855470839, 8.761543260320092012523303247857
