L(s) = 1 | + (1.5 − 2.59i)3-s + (7.90 + 13.6i)5-s + (−15.2 − 10.5i)7-s + (−4.5 − 7.79i)9-s + (−15.1 + 26.1i)11-s − 61.6·13-s + 47.4·15-s + (−28.0 + 48.6i)17-s + (−69.7 − 120. i)19-s + (−50.2 + 23.6i)21-s + (−4.07 − 7.05i)23-s + (−62.5 + 108. i)25-s − 27·27-s − 0.217·29-s + (88.2 − 152. i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.707 + 1.22i)5-s + (−0.820 − 0.570i)7-s + (−0.166 − 0.288i)9-s + (−0.414 + 0.718i)11-s − 1.31·13-s + 0.816·15-s + (−0.400 + 0.693i)17-s + (−0.841 − 1.45i)19-s + (−0.522 + 0.245i)21-s + (−0.0369 − 0.0639i)23-s + (−0.500 + 0.866i)25-s − 0.192·27-s − 0.00139·29-s + (0.511 − 0.885i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1560483599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1560483599\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 + (15.2 + 10.5i)T \) |
good | 5 | \( 1 + (-7.90 - 13.6i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15.1 - 26.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 61.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (28.0 - 48.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (69.7 + 120. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (4.07 + 7.05i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 0.217T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-88.2 + 152. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (105. + 182. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 293.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-241. - 418. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (10.2 - 17.7i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (115. - 200. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-419. - 726. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (312. - 540. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 227.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (21.5 - 37.2i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (154. + 267. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (572. + 991. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−11.50160104360333692246209287221, −10.32824614383813910071143196372, −10.01154115756769457526927381111, −8.851000240586810489717070103658, −7.33689832124780190316433399796, −6.95213234452487734195190424426, −6.04141531607069195598870678038, −4.46288248550870161095466522203, −2.91090521214840620545377601045, −2.17938858645135687486608757462,
0.04741685790870965054494493425, 2.00762851420218274584539190556, 3.30079031362791828306006312644, 4.84230665883305189467621165217, 5.48128807140860553403000941582, 6.64866987418591582904247713024, 8.244851714819679027756640098632, 8.833879742907643929739639042676, 9.798347713202680556522711621889, 10.25018191448629903730982079626