| L(s) = 1 | + 3·11-s + 5·13-s − 6·17-s − 19-s − 3·23-s + 6·29-s − 4·31-s − 11·37-s + 3·41-s − 10·43-s + 3·47-s + 3·53-s + 4·61-s − 4·67-s + 12·71-s − 4·73-s + 10·79-s − 12·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 1.38·13-s − 1.45·17-s − 0.229·19-s − 0.625·23-s + 1.11·29-s − 0.718·31-s − 1.80·37-s + 0.468·41-s − 1.52·43-s + 0.437·47-s + 0.412·53-s + 0.512·61-s − 0.488·67-s + 1.42·71-s − 0.468·73-s + 1.12·79-s − 1.31·83-s + 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.129482298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.129482298\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.23626936227590, −12.75278978012512, −12.18525864476586, −11.74143889272980, −11.29664397792979, −10.87570910860305, −10.36827188378967, −9.959327325520100, −9.197624138911365, −8.790533242280363, −8.586482763013441, −8.032904357385130, −7.273576148053180, −6.745974023389570, −6.421650260997986, −6.017925149128053, −5.272477157549287, −4.771709936812319, −4.078191400405089, −3.759603728287320, −3.232413525410968, −2.361258528828492, −1.820963142934844, −1.264196510586652, −0.4276999583267425,
0.4276999583267425, 1.264196510586652, 1.820963142934844, 2.361258528828492, 3.232413525410968, 3.759603728287320, 4.078191400405089, 4.771709936812319, 5.272477157549287, 6.017925149128053, 6.421650260997986, 6.745974023389570, 7.273576148053180, 8.032904357385130, 8.586482763013441, 8.790533242280363, 9.197624138911365, 9.959327325520100, 10.36827188378967, 10.87570910860305, 11.29664397792979, 11.74143889272980, 12.18525864476586, 12.75278978012512, 13.23626936227590
