L(s) = 1 | + 1.52i·2-s + 3·3-s + 5.67·4-s − 9.65i·5-s + 4.57i·6-s + 22.3i·7-s + 20.8i·8-s + 9·9-s + 14.7·10-s − 50.3i·11-s + 17.0·12-s + (−39.7 + 24.8i)13-s − 34.0·14-s − 28.9i·15-s + 13.6·16-s − 86.1·17-s + ⋯ |
L(s) = 1 | + 0.538i·2-s + 0.577·3-s + 0.709·4-s − 0.863i·5-s + 0.310i·6-s + 1.20i·7-s + 0.921i·8-s + 0.333·9-s + 0.465·10-s − 1.37i·11-s + 0.409·12-s + (−0.847 + 0.531i)13-s − 0.650·14-s − 0.498i·15-s + 0.213·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.61189 + 0.463421i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61189 + 0.463421i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 + (39.7 - 24.8i)T \) |
good | 2 | \( 1 - 1.52iT - 8T^{2} \) |
| 5 | \( 1 + 9.65iT - 125T^{2} \) |
| 7 | \( 1 - 22.3iT - 343T^{2} \) |
| 11 | \( 1 + 50.3iT - 1.33e3T^{2} \) |
| 17 | \( 1 + 86.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 14.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 154. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 265. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 67.5iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 686.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.9iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 329.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 768. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 264. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 771. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 514. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 527. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 74.2iT - 9.12e5T^{2} \) |
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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
−15.83823089385002916442091542516, −14.94577283398419799346550390836, −13.64459952880044262298616978040, −12.27284260297104871519878326131, −11.17801342567541892636590597802, −9.073746586965703148575828022714, −8.393218968189281304341396093993, −6.66188265347249156107358484717, −5.14729400854818857779589962431, −2.49482206629706922676557397874,
2.24565420657345228525944196038, 3.96217904350876123059012326389, 6.81759044521424932484784760918, 7.60769323165356513521241440575, 9.999201359348123166662267019749, 10.47200883765903349932721561952, 11.97843303298816643056632639017, 13.21491690717872269182295711793, 14.60059115800415823426312566289, 15.33654014751977186370184650853