L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 3.41i·11-s + (−2.02 + 2.02i)13-s + 1.00·14-s − 1.00·16-s + (−4.37 + 4.37i)17-s − 4.87i·19-s + (−2.41 − 2.41i)22-s + (−3.74 − 3.74i)23-s + 2.86i·26-s + (0.707 − 0.707i)28-s − 5.21·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.02i·11-s + (−0.561 + 0.561i)13-s + 0.267·14-s − 0.250·16-s + (−1.06 + 1.06i)17-s − 1.11i·19-s + (−0.514 − 0.514i)22-s + (−0.780 − 0.780i)23-s + 0.561i·26-s + (0.133 − 0.133i)28-s − 0.968·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4060791145\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4060791145\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 + 3.41iT - 11T^{2} \) |
| 13 | \( 1 + (2.02 - 2.02i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.37 - 4.37i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.87iT - 19T^{2} \) |
| 23 | \( 1 + (3.74 + 3.74i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 + 4.93T + 31T^{2} \) |
| 37 | \( 1 + (-6.87 - 6.87i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.77iT - 41T^{2} \) |
| 43 | \( 1 + (-0.174 + 0.174i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.43 + 6.43i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + (3.81 + 3.81i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (-2.51 + 2.51i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.98iT - 79T^{2} \) |
| 83 | \( 1 + (2.35 + 2.35i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 + (-5.64 - 5.64i)T + 97iT^{2} \) |
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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
−8.374593062732835602789702398551, −7.52590068478585856590374301353, −6.37221602567244998447168848948, −6.11819950997176178787267205660, −4.92489932920894712491710903124, −4.41479287538218029093758055017, −3.43368424519402483816116583796, −2.49896099946250439864773525395, −1.65167427656059872788116592166, −0.094847808979907444558789489842,
1.77421653523002986615034036355, 2.69069404878822044393461544350, 3.91266786046779354499690526490, 4.43098951412949537428018522903, 5.39770132957090487096312581966, 5.91652764237828878123399137645, 7.12479213755720623315989995198, 7.41582712146419793937548354774, 8.070091890593712631223575224369, 9.231638711087065080546929268853