| L(s) = 1 | + (0.707 + 0.707i)3-s + (−2 + i)5-s + (0.707 − 0.707i)7-s + 1.00i·9-s − 2.82i·11-s + (−3 + 3i)13-s + (−2.12 − 0.707i)15-s + (−1 − i)17-s − 2.82·19-s + 1.00·21-s + (−2.82 − 2.82i)23-s + (3 − 4i)25-s + (−0.707 + 0.707i)27-s − 8i·29-s + 8.48i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.894 + 0.447i)5-s + (0.267 − 0.267i)7-s + 0.333i·9-s − 0.852i·11-s + (−0.832 + 0.832i)13-s + (−0.547 − 0.182i)15-s + (−0.242 − 0.242i)17-s − 0.648·19-s + 0.218·21-s + (−0.589 − 0.589i)23-s + (0.600 − 0.800i)25-s + (−0.136 + 0.136i)27-s − 1.48i·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4244606018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4244606018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
| good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (2.82 + 2.82i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.65 + 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (5.65 - 5.65i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + (11.3 + 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (9 + 9i)T + 97iT^{2} \) |
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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
−8.776429402074930785367223685487, −8.519521352009588678714426406101, −7.44669612560142535315217384435, −6.92615637961207457908364999557, −5.84626572257785608322400598278, −4.62521039789881931661007378053, −4.08909877540454688640744356308, −3.13039099990642473581834132518, −2.10448591185899194652274340603, −0.14765755496898873332588825544,
1.51316190600882165920283186606, 2.63851282979309530073712403323, 3.72847616833573489123787147606, 4.64581722971009158000727706886, 5.40741298768931278536138154425, 6.61808141069863641463989716197, 7.45211504595261612944726026134, 7.982294756664006207359534580164, 8.665720916668257896514932545060, 9.508881292001852721800332934841
