L(s) = 1 | + (3.31 − 1.91i)2-s + (−0.144 + 5.19i)3-s + (3.33 − 5.77i)4-s + (2.5 + 4.33i)5-s + (9.46 + 17.5i)6-s + (8.94 + 16.2i)7-s + 5.08i·8-s + (−26.9 − 1.49i)9-s + (16.5 + 9.57i)10-s + (12.9 + 7.47i)11-s + (29.5 + 18.1i)12-s − 63.8i·13-s + (60.7 + 36.6i)14-s + (−22.8 + 12.3i)15-s + (36.4 + 63.0i)16-s + (11.0 − 19.0i)17-s + ⋯ |
L(s) = 1 | + (1.17 − 0.677i)2-s + (−0.0277 + 0.999i)3-s + (0.417 − 0.722i)4-s + (0.223 + 0.387i)5-s + (0.644 + 1.19i)6-s + (0.483 + 0.875i)7-s + 0.224i·8-s + (−0.998 − 0.0555i)9-s + (0.524 + 0.302i)10-s + (0.355 + 0.204i)11-s + (0.710 + 0.436i)12-s − 1.36i·13-s + (1.15 + 0.699i)14-s + (−0.393 + 0.212i)15-s + (0.569 + 0.985i)16-s + (0.156 − 0.271i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.76680 + 0.860693i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76680 + 0.860693i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.144 - 5.19i)T \) |
| 5 | \( 1 + (-2.5 - 4.33i)T \) |
| 7 | \( 1 + (-8.94 - 16.2i)T \) |
good | 2 | \( 1 + (-3.31 + 1.91i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-12.9 - 7.47i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 63.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-11.0 + 19.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-137. + 79.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (162. - 93.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 179. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-64.0 - 37.0i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (182. + 316. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 60.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.05 - 8.74i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-467. - 269. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-165. + 286. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (15.5 - 8.98i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (111. - 193. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 563. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-525. - 303. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-88.0 - 152. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 635.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (388. + 673. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 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
−13.55179278332240099577783103581, −12.06378492221693066408591473218, −11.56646767622913111702072860500, −10.45282567983586137849917110387, −9.384914882958093613600551144083, −7.982586780743620734404369837105, −5.74272349255936140702606594645, −5.13572022984785939832310275519, −3.64894712932605448290026508825, −2.54451696649818042970262843888,
1.38589635286185535403103399583, 3.76748213847925445189325834049, 5.12286411760325785221398457794, 6.33642651403879647825891690703, 7.20002431919896087922316229463, 8.360199139742321365381591799634, 9.983804332984922047868125303892, 11.71531997980968052440707764067, 12.30226381531756101353417378561, 13.72986771927652566252816965127