| L(s) = 1 | + 6.63i·5-s + 13.4i·7-s − 18.4·11-s − 18.9·25-s + 29.3i·29-s + 61.4i·31-s − 89.4·35-s − 132.·49-s − 105. i·53-s − 122. i·55-s + 117.·59-s + 143.·73-s − 248. i·77-s + 58i·79-s − 165.·83-s + ⋯ |
| L(s) = 1 | + 1.32i·5-s + 1.92i·7-s − 1.67·11-s − 0.758·25-s + 1.01i·29-s + 1.98i·31-s − 2.55·35-s − 2.71·49-s − 1.99i·53-s − 2.22i·55-s + 1.99·59-s + 1.96·73-s − 3.23i·77-s + 0.734i·79-s − 1.98·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9200338645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9200338645\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 6.63iT - 25T^{2} \) |
| 7 | \( 1 - 13.4iT - 49T^{2} \) |
| 11 | \( 1 + 18.4T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 29.3iT - 841T^{2} \) |
| 31 | \( 1 - 61.4iT - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 117.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 143.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 165.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 128.T + 9.40e3T^{2} \) |
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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
−9.731197487201931588337897663737, −8.661018832761980983147355736189, −8.218601955460601789227695407983, −7.12835143813770675967687164779, −6.49226646980777459113875287998, −5.42796900465872366107340795070, −5.12876994734724786786549407145, −3.33908590069716866549716745599, −2.75684779485349219653072775597, −2.04010529517000824752362310769,
0.27469055374492088567566089264, 0.987190930114747468350951789690, 2.39550814104313377710871444034, 3.81763647901023246962025327102, 4.47171985579669143614045262240, 5.18694342241594614285157440266, 6.13880432051641580775514142678, 7.39793098302283196904375365547, 7.77174818194062451736951000899, 8.475969012776861279342415559234
