| L(s) = 1 | − 2·2-s + 6.84·3-s + 4·4-s + 5·5-s − 13.6·6-s − 28.6·7-s − 8·8-s + 19.9·9-s − 10·10-s + 26.1·11-s + 27.3·12-s + 74.7·13-s + 57.2·14-s + 34.2·15-s + 16·16-s + 121.·17-s − 39.8·18-s − 6.41·19-s + 20·20-s − 195.·21-s − 52.2·22-s + 23·23-s − 54.7·24-s + 25·25-s − 149.·26-s − 48.5·27-s − 114.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.31·3-s + 0.5·4-s + 0.447·5-s − 0.932·6-s − 1.54·7-s − 0.353·8-s + 0.737·9-s − 0.316·10-s + 0.716·11-s + 0.659·12-s + 1.59·13-s + 1.09·14-s + 0.589·15-s + 0.250·16-s + 1.73·17-s − 0.521·18-s − 0.0775·19-s + 0.223·20-s − 2.03·21-s − 0.506·22-s + 0.208·23-s − 0.466·24-s + 0.200·25-s − 1.12·26-s − 0.346·27-s − 0.772·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.080178000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080178000\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 6.84T + 27T^{2} \) |
| 7 | \( 1 + 28.6T + 343T^{2} \) |
| 11 | \( 1 - 26.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.41T + 6.85e3T^{2} \) |
| 29 | \( 1 - 281.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 54.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 84.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 293.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 870.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 64.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 262.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 995.T + 9.12e5T^{2} \) |
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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
−11.75315893121473226174750436738, −10.26264198333893455563637506200, −9.691820733293265087394706184069, −8.856183241906944430418531394633, −8.141848119525796217840374533212, −6.78879163010938414608756163439, −5.97865185691510911930799936709, −3.60807349797765988781896815630, −2.94798691933596377241175459452, −1.23754612585730579670345808622,
1.23754612585730579670345808622, 2.94798691933596377241175459452, 3.60807349797765988781896815630, 5.97865185691510911930799936709, 6.78879163010938414608756163439, 8.141848119525796217840374533212, 8.856183241906944430418531394633, 9.691820733293265087394706184069, 10.26264198333893455563637506200, 11.75315893121473226174750436738
