L(s) = 1 | + 3i·3-s − 3.15·5-s + (−6.11 + 17.4i)7-s − 9·9-s − 60.9·11-s − 59.6·13-s − 9.46i·15-s + 21.4i·17-s − 95.5i·19-s + (−52.4 − 18.3i)21-s − 46.3i·23-s − 115.·25-s − 27i·27-s − 107. i·29-s + 94.2·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.282·5-s + (−0.329 + 0.944i)7-s − 0.333·9-s − 1.67·11-s − 1.27·13-s − 0.162i·15-s + 0.305i·17-s − 1.15i·19-s + (−0.545 − 0.190i)21-s − 0.419i·23-s − 0.920·25-s − 0.192i·27-s − 0.688i·29-s + 0.546·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7185280955\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7185280955\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (6.11 - 17.4i)T \) |
good | 5 | \( 1 + 3.15T + 125T^{2} \) |
| 11 | \( 1 + 60.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 95.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 46.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 107. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 94.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 131. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 283. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 373.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 298. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 468. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 160.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 409. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 930. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 442. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 190. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 829. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3iT - 9.12e5T^{2} \) |
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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
−9.405211260346677428756936758715, −8.358382564430911749758243502467, −7.81351023787433932420128952982, −6.76902023667995487668752589473, −5.67207684592268708960876723719, −5.06762253566244403080727899635, −4.21960036183887712320987492482, −2.75057489543255403495297896367, −2.47744776223618316059645582918, −0.28491435450897338326160332524,
0.55616295231438879635817211879, 2.03354863044959269555002218655, 2.99888122541283663838912459228, 4.06776851841886166727867797732, 5.13167767552702322423862683940, 5.89884272105022249005592723974, 7.13306707066761692973575633339, 7.56217191001870978439395762912, 8.092697242488144206846645531231, 9.334857486888488389401409724911