| L(s) = 1 | − 10.5·5-s − 55.2·11-s − 83.5·13-s − 95.2·17-s − 83.5·19-s − 165.·23-s − 12.9·25-s + 110.·29-s + 83.5·31-s + 78·37-s − 412.·41-s + 148·43-s + 465.·47-s + 110.·53-s + 584.·55-s − 550.·59-s + 584.·61-s + 883.·65-s − 260·67-s − 718.·71-s − 668.·73-s + 664·79-s − 126.·83-s + 1.00e3·85-s − 878.·89-s + 883.·95-s − 1.16e3·97-s + ⋯ |
| L(s) = 1 | − 0.946·5-s − 1.51·11-s − 1.78·13-s − 1.35·17-s − 1.00·19-s − 1.50·23-s − 0.103·25-s + 0.707·29-s + 0.483·31-s + 0.346·37-s − 1.57·41-s + 0.524·43-s + 1.44·47-s + 0.286·53-s + 1.43·55-s − 1.21·59-s + 1.22·61-s + 1.68·65-s − 0.474·67-s − 1.20·71-s − 1.07·73-s + 0.945·79-s − 0.167·83-s + 1.28·85-s − 1.04·89-s + 0.954·95-s − 1.22·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.04515378577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04515378577\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 10.5T + 125T^{2} \) |
| 11 | \( 1 + 55.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 83.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 83.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 78T + 5.06e4T^{2} \) |
| 41 | \( 1 + 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 148T + 7.95e4T^{2} \) |
| 47 | \( 1 - 465.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 110.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 550.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 584.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 260T + 3.00e5T^{2} \) |
| 71 | \( 1 + 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 668.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 664T + 4.93e5T^{2} \) |
| 83 | \( 1 + 126.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 878.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.16e3T + 9.12e5T^{2} \) |
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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
−8.737356467249622492359582750939, −8.064850213366088630906990168370, −7.49083756455809239200126436667, −6.70538494391781401515439459511, −5.62172487926904815225807346761, −4.63757819302238428076849174235, −4.17360186138485753276592018908, −2.75788660096163359864815712602, −2.16237842344201791916514034521, −0.091767982609116740211231269585,
0.091767982609116740211231269585, 2.16237842344201791916514034521, 2.75788660096163359864815712602, 4.17360186138485753276592018908, 4.63757819302238428076849174235, 5.62172487926904815225807346761, 6.70538494391781401515439459511, 7.49083756455809239200126436667, 8.064850213366088630906990168370, 8.737356467249622492359582750939
