| L(s) = 1 | + (−4.01 − 3.29i)3-s + (3.23 − 5.60i)5-s + (4.58 + 17.9i)7-s + (5.31 + 26.4i)9-s + (−2.23 − 3.86i)11-s + (−20.7 − 35.8i)13-s + (−31.4 + 11.8i)15-s + (−50.9 + 88.1i)17-s + (75.6 + 131. i)19-s + (40.6 − 87.2i)21-s + (60.2 − 104. i)23-s + (41.5 + 71.9i)25-s + (65.7 − 123. i)27-s + (114. − 198. i)29-s + 296.·31-s + ⋯ |
| L(s) = 1 | + (−0.773 − 0.633i)3-s + (0.289 − 0.501i)5-s + (0.247 + 0.968i)7-s + (0.196 + 0.980i)9-s + (−0.0611 − 0.105i)11-s + (−0.441 − 0.765i)13-s + (−0.541 + 0.204i)15-s + (−0.726 + 1.25i)17-s + (0.913 + 1.58i)19-s + (0.422 − 0.906i)21-s + (0.546 − 0.946i)23-s + (0.332 + 0.575i)25-s + (0.468 − 0.883i)27-s + (0.735 − 1.27i)29-s + 1.71·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.381375797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381375797\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.01 + 3.29i)T \) |
| 7 | \( 1 + (-4.58 - 17.9i)T \) |
| good | 5 | \( 1 + (-3.23 + 5.60i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (2.23 + 3.86i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (20.7 + 35.8i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (50.9 - 88.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-75.6 - 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-60.2 + 104. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-114. + 198. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 - 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (89.8 + 155. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-242. - 419. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (57.7 - 99.9i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 507.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (57.0 - 98.7i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + 297.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 10.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 322.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 799.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (417. - 723. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + 933.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-625. + 1.08e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (248. + 430. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (735. - 1.27e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.86243974065875085692225760510, −10.77008133970856809986050238430, −9.838611562619231066567477349252, −8.489336774730175961491888792984, −7.83014589926216048979301410891, −6.29836297290244215645664266864, −5.66567031335154044835195094681, −4.60651697723849983582679241656, −2.52893639610603591410384397243, −1.13393218587757746332854195068,
0.74833043620902575067546445510, 2.89811601869194041415893738163, 4.44414673993908795639485549879, 5.14480519537782329314301618258, 6.81600936063415035391786499788, 7.09709507849398236541707357976, 8.987973198072347889773773060455, 9.784209922231906247914026792372, 10.72281945594993301030393762461, 11.35863095106871409559390352726
