L(s) = 1 | + 1.41·2-s + 1.00·4-s + 5-s − 1.41·7-s + 1.41·10-s − 11-s + 1.41·13-s − 2.00·14-s − 0.999·16-s − 1.41·17-s + 1.00·20-s − 1.41·22-s + 25-s + 2.00·26-s − 1.41·28-s − 1.41·32-s − 2.00·34-s − 1.41·35-s + 1.41·43-s − 1.00·44-s + 1.00·49-s + 1.41·50-s + 1.41·52-s − 55-s − 1.00·64-s + 1.41·65-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s + 5-s − 1.41·7-s + 1.41·10-s − 11-s + 1.41·13-s − 2.00·14-s − 0.999·16-s − 1.41·17-s + 1.00·20-s − 1.41·22-s + 25-s + 2.00·26-s − 1.41·28-s − 1.41·32-s − 2.00·34-s − 1.41·35-s + 1.41·43-s − 1.00·44-s + 1.00·49-s + 1.41·50-s + 1.41·52-s − 55-s − 1.00·64-s + 1.41·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.651781096\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651781096\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 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 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.41T + T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.20527720293515460316132033344, −10.47715785806306955454097860720, −9.413808162322701135677632800148, −8.674904304332766192542308775724, −6.96188246317849590672782469095, −6.18230644516051652951258386661, −5.68195058062788999199735320572, −4.46091705236518078993507275027, −3.32019965927162093848879516456, −2.40520655674812140903479631954,
2.40520655674812140903479631954, 3.32019965927162093848879516456, 4.46091705236518078993507275027, 5.68195058062788999199735320572, 6.18230644516051652951258386661, 6.96188246317849590672782469095, 8.674904304332766192542308775724, 9.413808162322701135677632800148, 10.47715785806306955454097860720, 11.20527720293515460316132033344