| L(s) = 1 | − 22·5-s − 27·9-s + 18·13-s + 94·17-s + 359·25-s − 130·29-s + 214·37-s + 230·41-s + 594·45-s + 518·53-s − 830·61-s − 396·65-s − 1.09e3·73-s + 729·81-s − 2.06e3·85-s + 1.67e3·89-s − 594·97-s − 598·101-s − 1.74e3·109-s + 2.00e3·113-s − 486·117-s + ⋯ |
| L(s) = 1 | − 1.96·5-s − 9-s + 0.384·13-s + 1.34·17-s + 2.87·25-s − 0.832·29-s + 0.950·37-s + 0.876·41-s + 1.96·45-s + 1.34·53-s − 1.74·61-s − 0.755·65-s − 1.76·73-s + 81-s − 2.63·85-s + 1.98·89-s − 0.621·97-s − 0.589·101-s − 1.53·109-s + 1.66·113-s − 0.384·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + 22 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 - 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 518 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−8.541479273565226420997584150244, −7.77290942806103037000343903714, −7.43080982229133945446795273805, −6.21788495201735933009266073466, −5.32116043328747685271201496409, −4.27699966036892734845975951183, −3.53949045812552346374299187657, −2.81119857647008874911943997274, −0.997388644218494631556116533723, 0,
0.997388644218494631556116533723, 2.81119857647008874911943997274, 3.53949045812552346374299187657, 4.27699966036892734845975951183, 5.32116043328747685271201496409, 6.21788495201735933009266073466, 7.43080982229133945446795273805, 7.77290942806103037000343903714, 8.541479273565226420997584150244
