L(s) = 1 | + (1.73 + 0.564i)2-s + (1.89 + 1.37i)4-s + (0.587 − 0.809i)7-s + (1.43 + 1.97i)8-s + (−0.309 + 0.951i)9-s + (−0.978 − 0.207i)11-s + (1.47 − 1.07i)14-s + (0.657 + 2.02i)16-s + (−1.07 + 1.47i)18-s + (−1.58 − 0.913i)22-s + 0.209i·23-s + (2.22 − 0.722i)28-s + (−1.58 − 1.14i)29-s + 1.44i·32-s + (−1.89 + 1.37i)36-s + (1.07 − 1.47i)37-s + ⋯ |
L(s) = 1 | + (1.73 + 0.564i)2-s + (1.89 + 1.37i)4-s + (0.587 − 0.809i)7-s + (1.43 + 1.97i)8-s + (−0.309 + 0.951i)9-s + (−0.978 − 0.207i)11-s + (1.47 − 1.07i)14-s + (0.657 + 2.02i)16-s + (−1.07 + 1.47i)18-s + (−1.58 − 0.913i)22-s + 0.209i·23-s + (2.22 − 0.722i)28-s + (−1.58 − 1.14i)29-s + 1.44i·32-s + (−1.89 + 1.37i)36-s + (1.07 − 1.47i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.081274422\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.081274422\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-0.587 + 0.809i)T \) |
| 11 | \( 1 + (0.978 + 0.207i)T \) |
good | 2 | \( 1 + (-1.73 - 0.564i)T + (0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - 0.209iT - T^{2} \) |
| 29 | \( 1 + (1.58 + 1.14i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.07 + 1.47i)T + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 1.95iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 1.33iT - T^{2} \) |
| 71 | \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.413 - 1.27i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
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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
−9.565573908369971118615033437682, −8.050787770801778873349041362928, −7.80841366505452868529395436789, −7.11886024847738284655650533169, −5.98051167550570332304037136841, −5.45091122817873632303688626323, −4.64545440392977050167704905473, −4.02392931139826292545670122743, −2.93326725524843333581065944847, −2.04014203611695339093559670541,
1.67680254133498488568012678783, 2.65550097426168998035008162250, 3.38169392908944088262061741264, 4.37859341780840517411889648550, 5.23943973779034574364222728582, 5.69906048202136760575824640641, 6.54158563694612814384119405863, 7.45235242619962003870173683508, 8.515213268600562484106743402199, 9.408402175461635104168845408621