| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 2.64i·7-s − 2.82·8-s − 5.31i·11-s + 0.354i·13-s + 3.74i·14-s + 4.00·16-s + 31.5·17-s + 28.2·19-s + 7.52i·22-s + 25.0·23-s − 0.500i·26-s − 5.29i·28-s + 3.16i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.377i·7-s − 0.353·8-s − 0.483i·11-s + 0.0272i·13-s + 0.267i·14-s + 0.250·16-s + 1.85·17-s + 1.48·19-s + 0.341i·22-s + 1.08·23-s − 0.0192i·26-s − 0.188i·28-s + 0.109i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.782550113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.782550113\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 11 | \( 1 + 5.31iT - 121T^{2} \) |
| 13 | \( 1 - 0.354iT - 169T^{2} \) |
| 17 | \( 1 - 31.5T + 289T^{2} \) |
| 19 | \( 1 - 28.2T + 361T^{2} \) |
| 23 | \( 1 - 25.0T + 529T^{2} \) |
| 29 | \( 1 - 3.16iT - 841T^{2} \) |
| 31 | \( 1 - 25.1T + 961T^{2} \) |
| 37 | \( 1 - 41.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 70.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 73.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 46.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 41.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 56.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 86.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 128.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 142. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 133. iT - 9.40e3T^{2} \) |
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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
−8.306881053639986650891744769347, −7.895998552187653491109724520093, −7.14768701042042172325749233030, −6.37360617403411443054393176847, −5.48816882520209144568514961835, −4.77387427962322818707621442468, −3.33121897963001035359840884486, −3.05144573067151943947818419838, −1.43342259442840955446269244425, −0.805863160830960253253723514124,
0.76420683593828782456420581314, 1.66327761599364435525058310590, 2.88662745739742284717686001240, 3.49327217048652053679231088557, 4.88572035724229832209687134699, 5.49353386018887614290759129559, 6.35350477182419943129449856733, 7.31313840588289521846868098735, 7.72217085503246889605408239796, 8.524733308930791805796879040259
