| L(s) = 1 | − 1.63·2-s − 5.33·4-s + 12.3·5-s + 21.7·8-s − 20.2·10-s − 29.0·11-s − 52.9·13-s + 7.18·16-s − 122.·17-s + 141.·19-s − 66.0·20-s + 47.3·22-s + 60.2·23-s + 28.2·25-s + 86.4·26-s − 126.·29-s + 150.·31-s − 185.·32-s + 199.·34-s − 341.·37-s − 230.·38-s + 269.·40-s + 292.·41-s + 290.·43-s + 154.·44-s − 98.3·46-s + 284.·47-s + ⋯ |
| L(s) = 1 | − 0.576·2-s − 0.667·4-s + 1.10·5-s + 0.961·8-s − 0.638·10-s − 0.795·11-s − 1.13·13-s + 0.112·16-s − 1.74·17-s + 1.70·19-s − 0.738·20-s + 0.458·22-s + 0.546·23-s + 0.225·25-s + 0.652·26-s − 0.813·29-s + 0.873·31-s − 1.02·32-s + 1.00·34-s − 1.51·37-s − 0.982·38-s + 1.06·40-s + 1.11·41-s + 1.03·43-s + 0.530·44-s − 0.315·46-s + 0.881·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.170251342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170251342\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.63T + 8T^{2} \) |
| 5 | \( 1 - 12.3T + 125T^{2} \) |
| 11 | \( 1 + 29.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 60.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 150.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 292.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 269.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 239.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 712.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 270.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 146.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 652.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 35.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 805.T + 9.12e5T^{2} \) |
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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
−9.258253384852841714954852901159, −8.760975053492387201892472556998, −7.59334869017956356302516858660, −7.08253696310729198901740202132, −5.74407357763876428622812804977, −5.14262597096247036350807784449, −4.30550543225456266162042882036, −2.80425838869806080043994210985, −1.89891939543967985502763485048, −0.58934184109266061182029721433,
0.58934184109266061182029721433, 1.89891939543967985502763485048, 2.80425838869806080043994210985, 4.30550543225456266162042882036, 5.14262597096247036350807784449, 5.74407357763876428622812804977, 7.08253696310729198901740202132, 7.59334869017956356302516858660, 8.760975053492387201892472556998, 9.258253384852841714954852901159
