L(s) = 1 | + (−0.0922 + 0.995i)2-s + (−0.982 − 0.183i)4-s + (−1.85 + 0.526i)5-s + (0.273 − 0.961i)8-s + (−0.445 − 0.895i)9-s + (−0.353 − 1.89i)10-s + (0.243 − 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (1.91 − 0.177i)20-s + (2.29 − 1.42i)25-s + (0.831 + 0.322i)26-s + (0.328 − 1.75i)29-s + (−0.445 + 0.895i)32-s + ⋯ |
L(s) = 1 | + (−0.0922 + 0.995i)2-s + (−0.982 − 0.183i)4-s + (−1.85 + 0.526i)5-s + (0.273 − 0.961i)8-s + (−0.445 − 0.895i)9-s + (−0.353 − 1.89i)10-s + (0.243 − 0.857i)13-s + (0.932 + 0.361i)16-s + (0.932 + 0.361i)17-s + (0.932 − 0.361i)18-s + (1.91 − 0.177i)20-s + (2.29 − 1.42i)25-s + (0.831 + 0.322i)26-s + (0.328 − 1.75i)29-s + (−0.445 + 0.895i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4661969526\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4661969526\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0922 - 0.995i)T \) |
| 17 | \( 1 + (-0.932 - 0.361i)T \) |
good | 3 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 5 | \( 1 + (1.85 - 0.526i)T + (0.850 - 0.526i)T^{2} \) |
| 7 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 11 | \( 1 + (0.932 + 0.361i)T^{2} \) |
| 13 | \( 1 + (-0.243 + 0.857i)T + (-0.850 - 0.526i)T^{2} \) |
| 19 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 23 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 29 | \( 1 + (-0.328 + 1.75i)T + (-0.932 - 0.361i)T^{2} \) |
| 31 | \( 1 + (-0.850 - 0.526i)T^{2} \) |
| 37 | \( 1 + (1.34 + 1.47i)T + (-0.0922 + 0.995i)T^{2} \) |
| 41 | \( 1 + (1.01 - 1.63i)T + (-0.445 - 0.895i)T^{2} \) |
| 43 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 47 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 53 | \( 1 + (0.537 + 1.07i)T + (-0.602 + 0.798i)T^{2} \) |
| 59 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 61 | \( 1 + (0.840 + 0.634i)T + (0.273 + 0.961i)T^{2} \) |
| 67 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 71 | \( 1 + (-0.602 + 0.798i)T^{2} \) |
| 73 | \( 1 + (-0.646 + 1.66i)T + (-0.739 - 0.673i)T^{2} \) |
| 79 | \( 1 + (-0.982 + 0.183i)T^{2} \) |
| 83 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 89 | \( 1 + (-0.0505 - 0.177i)T + (-0.850 + 0.526i)T^{2} \) |
| 97 | \( 1 + (-0.328 + 0.163i)T + (0.602 - 0.798i)T^{2} \) |
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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
−9.873246171587216475576477074977, −8.753288147983193481833107171378, −8.089490342095046976836439803237, −7.63166794000915641726973520868, −6.70828272387722149743137454624, −5.94998952808299838548447543368, −4.81539506095129577271570619713, −3.71927521467983542560920003063, −3.31193864998621159621082374776, −0.48680214664612358280771602990,
1.37454153953959900617315380185, 3.00326244258350917579313255972, 3.75530452242500841011224012101, 4.70739435954526970234842868267, 5.28512380411110578804626344642, 7.07969855168377677610949515370, 7.79638482414143238962295337206, 8.613324336420931144317678695722, 8.965349875021027035083787038466, 10.32702603108617945715269490342