| L(s) = 1 | + (1 − 1.73i)2-s − 3-s + (−1.99 − 3.46i)4-s + 5.19i·5-s + (−1 + 1.73i)6-s − 7.99·8-s − 8·9-s + (9 + 5.19i)10-s + 17·11-s + (1.99 + 3.46i)12-s + 13.8i·13-s − 5.19i·15-s + (−8 + 13.8i)16-s + 25·17-s + (−8 + 13.8i)18-s + 7·19-s + ⋯ |
| L(s) = 1 | + (0.5 − 0.866i)2-s − 0.333·3-s + (−0.499 − 0.866i)4-s + 1.03i·5-s + (−0.166 + 0.288i)6-s − 0.999·8-s − 0.888·9-s + (0.900 + 0.519i)10-s + 1.54·11-s + (0.166 + 0.288i)12-s + 1.06i·13-s − 0.346i·15-s + (−0.5 + 0.866i)16-s + 1.47·17-s + (−0.444 + 0.769i)18-s + 0.368·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73968\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + 9T^{2} \) |
| 5 | \( 1 - 5.19iT - 25T^{2} \) |
| 11 | \( 1 - 17T + 121T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 - 25T + 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 - 5.19iT - 529T^{2} \) |
| 29 | \( 1 - 13.8iT - 841T^{2} \) |
| 31 | \( 1 - 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 8.66iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 26T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 91.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 55T + 3.48e3T^{2} \) |
| 61 | \( 1 + 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 17T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 119T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 110T + 6.88e3T^{2} \) |
| 89 | \( 1 + 71T + 7.92e3T^{2} \) |
| 97 | \( 1 - 22T + 9.40e3T^{2} \) |
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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
−11.43789260012056239983752353882, −10.37895187602942035116797433515, −9.529740070795587725176968247395, −8.628411408707995927998886263845, −6.95880904313581663838776603092, −6.24691185513986460217230995642, −5.18486713674216356361697558968, −3.79514204272250114156908163198, −2.97714212593099181607820601663, −1.39289197047827418099796665046,
0.78051910570844689253260054346, 3.21196555244494823570260519374, 4.36246463659276340983553499780, 5.51658998963175856723343852319, 5.96402390454118710490862691215, 7.30494497934845861596838027199, 8.314146134781319181801148941836, 8.951427502160742520197591491177, 9.952048856205152647604821499509, 11.53763711716002954420867499198
