| L(s) = 1 | + 4i·2-s + (12.3 + 9.52i)3-s − 16·4-s − 12.2·5-s + (−38.1 + 49.3i)6-s − 64i·8-s + (61.5 + 235. i)9-s − 48.8i·10-s + 198. i·11-s + (−197. − 152. i)12-s + 369. i·13-s + (−150. − 116. i)15-s + 256·16-s + 896.·17-s + (−940. + 246. i)18-s + 1.56e3i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.791 + 0.611i)3-s − 0.5·4-s − 0.218·5-s + (−0.432 + 0.559i)6-s − 0.353i·8-s + (0.253 + 0.967i)9-s − 0.154i·10-s + 0.494i·11-s + (−0.395 − 0.305i)12-s + 0.606i·13-s + (−0.172 − 0.133i)15-s + 0.250·16-s + 0.752·17-s + (−0.684 + 0.179i)18-s + 0.993i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.507469300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.507469300\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 + (-12.3 - 9.52i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 12.2T + 3.12e3T^{2} \) |
| 11 | \( 1 - 198. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 369. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 896.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 793. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 496. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.72e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.46e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.69e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 6.61e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.55e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.44e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 3.30e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.26e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.48e4iT - 8.58e9T^{2} \) |
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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
−11.44618693392962602785638540054, −10.06392869173096888858793719138, −9.646074494639052929727728255564, −8.418556204769572513639815656361, −7.83894332520388345279317265610, −6.73269375603385134327710905109, −5.40183721022034511777578469452, −4.33572359669992332368494719851, −3.40460250498688053615316063612, −1.79093661003629559881893998920,
0.35614375049210587406643120583, 1.57801649151986193240858033971, 2.90208410644541028529495577971, 3.68117666176465554429877731445, 5.18668241627423764910503638508, 6.55558640530163941385596390713, 7.76253641320282931167160896033, 8.453227673955830409935517620819, 9.455065331623830950572725998693, 10.32845332942050320318111892741
