| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 4·5-s − 2·6-s − 7-s + 9-s − 8·10-s − 2·12-s − 2·14-s + 4·15-s − 4·16-s + 4·17-s + 2·18-s + 3·19-s − 8·20-s + 21-s + 2·23-s + 11·25-s − 27-s − 2·28-s + 6·29-s + 8·30-s + 5·31-s − 8·32-s + 8·34-s + 4·35-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s − 2.52·10-s − 0.577·12-s − 0.534·14-s + 1.03·15-s − 16-s + 0.970·17-s + 0.471·18-s + 0.688·19-s − 1.78·20-s + 0.218·21-s + 0.417·23-s + 11/5·25-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.46·30-s + 0.898·31-s − 1.41·32-s + 1.37·34-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.616543821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.616543821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.31671423761629, −13.80993069425046, −13.18048417033380, −12.60502875108195, −12.22591284942235, −12.01270309988031, −11.46710135066986, −11.03824406659608, −10.49629465304877, −9.756600778451342, −9.195114835844325, −8.341980187615726, −8.019198964333724, −7.284721386045800, −6.839955377275199, −6.418811317294634, −5.525492847944440, −5.260093207276821, −4.542360477076675, −4.101675760839937, −3.628287516642843, −3.035371312564239, −2.562092663591335, −1.121207936600801, −0.5245534459667503,
0.5245534459667503, 1.121207936600801, 2.562092663591335, 3.035371312564239, 3.628287516642843, 4.101675760839937, 4.542360477076675, 5.260093207276821, 5.525492847944440, 6.418811317294634, 6.839955377275199, 7.284721386045800, 8.019198964333724, 8.341980187615726, 9.195114835844325, 9.756600778451342, 10.49629465304877, 11.03824406659608, 11.46710135066986, 12.01270309988031, 12.22591284942235, 12.60502875108195, 13.18048417033380, 13.80993069425046, 14.31671423761629
