L(s) = 1 | − 4·7-s + 5·13-s − 19-s − 7·31-s + 11·37-s + 5·43-s + 9·49-s + 14·61-s − 16·67-s − 10·73-s − 4·79-s − 20·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.51·7-s + 1.38·13-s − 0.229·19-s − 1.25·31-s + 1.80·37-s + 0.762·43-s + 9/7·49-s + 1.79·61-s − 1.95·67-s − 1.17·73-s − 0.450·79-s − 2.09·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.771920820\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.771920820\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.89106552150639, −12.59654691721510, −11.76250712911570, −11.52335717268091, −10.93661315426660, −10.42924730532819, −10.14692558821259, −9.434662275622893, −9.168906124301397, −8.772600849733712, −8.165003291962449, −7.612684024501994, −7.111714629348313, −6.604814279492519, −6.082056693107612, −5.877419699307997, −5.314912273465133, −4.400857889736928, −4.057963185958687, −3.508480262635310, −3.049792242171909, −2.513906364404203, −1.770748977230878, −1.030365211715883, −0.4076774511828008,
0.4076774511828008, 1.030365211715883, 1.770748977230878, 2.513906364404203, 3.049792242171909, 3.508480262635310, 4.057963185958687, 4.400857889736928, 5.314912273465133, 5.877419699307997, 6.082056693107612, 6.604814279492519, 7.111714629348313, 7.612684024501994, 8.165003291962449, 8.772600849733712, 9.168906124301397, 9.434662275622893, 10.14692558821259, 10.42924730532819, 10.93661315426660, 11.52335717268091, 11.76250712911570, 12.59654691721510, 12.89106552150639