L(s) = 1 | + 2-s − 3-s + 4-s + 1.41·5-s − 6-s + 8-s + 9-s + 1.41·10-s − 12-s − 1.41·15-s + 16-s − 1.41·17-s + 18-s + 1.41·20-s − 24-s + 1.00·25-s − 27-s − 1.41·30-s + 32-s − 1.41·34-s + 36-s + 1.41·40-s − 1.41·41-s − 1.41·43-s + 1.41·45-s − 48-s + 49-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s + 1.41·5-s − 6-s + 8-s + 9-s + 1.41·10-s − 12-s − 1.41·15-s + 16-s − 1.41·17-s + 18-s + 1.41·20-s − 24-s + 1.00·25-s − 27-s − 1.41·30-s + 32-s − 1.41·34-s + 36-s + 1.41·40-s − 1.41·41-s − 1.41·43-s + 1.41·45-s − 48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.050433302\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.050433302\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.684898670507145028234211755429, −8.600848142510647917293385536955, −7.34801173715297184623907864971, −6.46570180781856361821142361865, −6.27053933013637061074273984398, −5.22257374137427325120718718893, −4.84362999309174022357318756433, −3.71699390632672478748566771544, −2.36626245041743144002089961715, −1.58147370683251682716217929803,
1.58147370683251682716217929803, 2.36626245041743144002089961715, 3.71699390632672478748566771544, 4.84362999309174022357318756433, 5.22257374137427325120718718893, 6.27053933013637061074273984398, 6.46570180781856361821142361865, 7.34801173715297184623907864971, 8.600848142510647917293385536955, 9.684898670507145028234211755429