| L(s) = 1 | + 5.65·2-s + 15.5·3-s + 32.0·4-s + 56.1i·5-s + 88.1·6-s − 504. i·7-s + 181.·8-s + 243·9-s + 317. i·10-s − 2.00e3i·11-s + 498.·12-s − 3.17e3·13-s − 2.85e3i·14-s + 875. i·15-s + 1.02e3·16-s + 557. i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.500·4-s + 0.449i·5-s + 0.408·6-s − 1.46i·7-s + 0.353·8-s + 0.333·9-s + 0.317i·10-s − 1.50i·11-s + 0.288·12-s − 1.44·13-s − 1.03i·14-s + 0.259i·15-s + 0.250·16-s + 0.113i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.287628538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.287628538\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 - 15.5T \) |
| 23 | \( 1 + (1.58e3 + 1.20e4i)T \) |
| good | 5 | \( 1 - 56.1iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 504. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 2.00e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.17e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 557. iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 872. iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 1.75e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 5.93e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 1.72e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 9.13e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 7.95e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 7.13e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.76e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.28e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 5.65e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 1.45e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.05e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.56e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.77e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.46e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 6.33e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.13e6iT - 8.32e11T^{2} \) |
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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
−11.89346742383644216570797075427, −10.72209238725521597852913412347, −10.07906885784075635988395113183, −8.454047268808174434557402507709, −7.35111024670524703688197686816, −6.49363889382707254757813665583, −4.83883885654377160501941669422, −3.66981751925811210453000866350, −2.62462239251456409693284420345, −0.71027301839463513578842234430,
1.89196270152125427586000183869, 2.80009607055482555177490863250, 4.55300039035134451027815282151, 5.35322317942030357524789012146, 6.88062287745963599539694268367, 7.991778834830262133609361522334, 9.260374428286059950955423042083, 9.994639433618234946952802968522, 11.83424850480312006604175761693, 12.30289414982725400937960472280
