| L(s) = 1 | + (0.325 − 1.70i)3-s + (−1.34 − 1.78i)5-s + (−1.53 − 2.15i)7-s + (−2.78 − 1.10i)9-s − 5.00i·11-s + 1.62·13-s + (−3.47 + 1.70i)15-s − 4.73i·17-s − 2.13i·19-s + (−4.16 + 1.91i)21-s + 9.46·23-s + (−1.38 + 4.80i)25-s + (−2.79 + 4.38i)27-s + 5.96i·29-s − 4.38i·31-s + ⋯ |
| L(s) = 1 | + (0.187 − 0.982i)3-s + (−0.601 − 0.798i)5-s + (−0.582 − 0.813i)7-s + (−0.929 − 0.369i)9-s − 1.51i·11-s + 0.451·13-s + (−0.897 + 0.440i)15-s − 1.14i·17-s − 0.490i·19-s + (−0.908 + 0.418i)21-s + 1.97·23-s + (−0.276 + 0.961i)25-s + (−0.537 + 0.843i)27-s + 1.10i·29-s − 0.787i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.136228675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.136228675\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.325 + 1.70i)T \) |
| 5 | \( 1 + (1.34 + 1.78i)T \) |
| 7 | \( 1 + (1.53 + 2.15i)T \) |
| good | 11 | \( 1 + 5.00iT - 11T^{2} \) |
| 13 | \( 1 - 1.62T + 13T^{2} \) |
| 17 | \( 1 + 4.73iT - 17T^{2} \) |
| 19 | \( 1 + 2.13iT - 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 - 5.96iT - 29T^{2} \) |
| 31 | \( 1 + 4.38iT - 31T^{2} \) |
| 37 | \( 1 - 3.75iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.74iT - 43T^{2} \) |
| 47 | \( 1 + 5.64iT - 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 2.09iT - 61T^{2} \) |
| 67 | \( 1 - 6.20iT - 67T^{2} \) |
| 71 | \( 1 + 8.65iT - 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 1.06T + 79T^{2} \) |
| 83 | \( 1 - 5.96iT - 83T^{2} \) |
| 89 | \( 1 + 0.187T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.771256553221497192228344913858, −8.179488980398890421584467975165, −7.16910081239408096338602352147, −6.78913722750472902109257060353, −5.66502019372327927305904703306, −4.83576969418077299314569200163, −3.49094358389750484711024963387, −3.00176238503095590552236241335, −1.16109708423168588438428122587, −0.47950426169922323128835000061,
2.12283322244406771314130860895, 3.11982008529284111295141497718, 3.85237273672218386426464687427, 4.74947285136785701460128595686, 5.72161253010781075981379131118, 6.61728363998878970807271609486, 7.40945540179971537629110994608, 8.455649147968864105597942816108, 8.985800584157394515692013882281, 10.01531766056911142698484489094
