L(s) = 1 | + (4.5 − 7.79i)3-s + (21.8 + 37.9i)5-s + (65.0 − 112. i)7-s + (−40.5 − 70.1i)9-s + (167. − 289. i)11-s + 51.3·13-s + 393.·15-s + (−45.1 + 78.2i)17-s + (−127. − 220. i)19-s + (−580. − 1.01e3i)21-s + (−208. − 360. i)23-s + (604. − 1.04e3i)25-s − 729·27-s − 3.10e3·29-s + (−37.3 + 64.6i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.391 + 0.678i)5-s + (0.502 − 0.864i)7-s + (−0.166 − 0.288i)9-s + (0.416 − 0.720i)11-s + 0.0842·13-s + 0.452·15-s + (−0.0379 + 0.0656i)17-s + (−0.0809 − 0.140i)19-s + (−0.287 − 0.500i)21-s + (−0.0820 − 0.142i)23-s + (0.193 − 0.335i)25-s − 0.192·27-s − 0.685·29-s + (−0.00697 + 0.0120i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0657 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0657 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.405006763\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.405006763\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 + 7.79i)T \) |
| 7 | \( 1 + (-65.0 + 112. i)T \) |
good | 5 | \( 1 + (-21.8 - 37.9i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-167. + 289. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 51.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + (45.1 - 78.2i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (127. + 220. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (208. + 360. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.10e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (37.3 - 64.6i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.46e3 + 4.26e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 3.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.72e3 - 1.16e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.13e3 + 5.42e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (561. - 972. i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.53e4 + 2.65e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.06e4 + 5.30e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.11e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + (5.20e3 - 9.01e3i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.16e4 + 3.74e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 2.50e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (4.32e4 + 7.48e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 5.93e4T + 8.58e9T^{2} \) |
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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
−10.69126871938008487642943968274, −9.544037679032589397557794901085, −8.505190110186059357281216358574, −7.54508282415293569312326077355, −6.71539262356040651809108300113, −5.78243688604441288487689857289, −4.27048490396791631360801044204, −3.13499006965590673779181572011, −1.85833143735949168971526975790, −0.60570501136338708884121567735,
1.38604470324550640021783275840, 2.48078712095108817677305615723, 4.00871956708063414696487661606, 5.03593944033276060812121773299, 5.82188334231095031094095949246, 7.25088550161684459992718086179, 8.431267809681161061236490924176, 9.109608212445446766635273952872, 9.809061816230231719923855516715, 10.94892096070861403996508985152