| L(s) = 1 | + 7-s − 4·11-s + 5·13-s + 3·17-s + 2·19-s − 23-s + 9·29-s + 3·31-s + 10·37-s + 5·41-s − 9·43-s + 6·47-s + 49-s + 3·53-s − 3·59-s + 5·61-s − 8·67-s − 4·73-s − 4·77-s − 4·79-s + 9·83-s + 6·89-s + 5·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s + 1.38·13-s + 0.727·17-s + 0.458·19-s − 0.208·23-s + 1.67·29-s + 0.538·31-s + 1.64·37-s + 0.780·41-s − 1.37·43-s + 0.875·47-s + 1/7·49-s + 0.412·53-s − 0.390·59-s + 0.640·61-s − 0.977·67-s − 0.468·73-s − 0.455·77-s − 0.450·79-s + 0.987·83-s + 0.635·89-s + 0.524·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.094822071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.094822071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.45130325173759, −13.93010317217848, −13.51302430035574, −13.06151089503392, −12.52658781738783, −11.80359222992423, −11.53945013595750, −10.83984045308947, −10.33377854535983, −10.06053797546060, −9.270385300061378, −8.636139952555295, −8.190189159943323, −7.758907183416815, −7.236289453871815, −6.306942803695695, −6.034824606475456, −5.348165004750883, −4.760894983995019, −4.206094349259609, −3.384604502939316, −2.873755425964757, −2.189704326553880, −1.212369899088562, −0.7013434371935557,
0.7013434371935557, 1.212369899088562, 2.189704326553880, 2.873755425964757, 3.384604502939316, 4.206094349259609, 4.760894983995019, 5.348165004750883, 6.034824606475456, 6.306942803695695, 7.236289453871815, 7.758907183416815, 8.190189159943323, 8.636139952555295, 9.270385300061378, 10.06053797546060, 10.33377854535983, 10.83984045308947, 11.53945013595750, 11.80359222992423, 12.52658781738783, 13.06151089503392, 13.51302430035574, 13.93010317217848, 14.45130325173759
