| L(s) = 1 | − 7.40·5-s + 9.47i·7-s − 6.77i·11-s + 14.4·13-s + 17.8·17-s + 5.19i·19-s − 24.9i·23-s + 29.8·25-s − 29.6·29-s − 17.1i·31-s − 70.1i·35-s − 6.41·37-s + 8.64·41-s + 50.1i·43-s − 52.0i·47-s + ⋯ |
| L(s) = 1 | − 1.48·5-s + 1.35i·7-s − 0.615i·11-s + 1.10·13-s + 1.05·17-s + 0.273i·19-s − 1.08i·23-s + 1.19·25-s − 1.02·29-s − 0.552i·31-s − 2.00i·35-s − 0.173·37-s + 0.210·41-s + 1.16i·43-s − 1.10i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.190968371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.190968371\) |
| \(L(2)\) |
|
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 | \( 1 + 7.40T + 25T^{2} \) |
| 7 | \( 1 - 9.47iT - 49T^{2} \) |
| 11 | \( 1 + 6.77iT - 121T^{2} \) |
| 13 | \( 1 - 14.4T + 169T^{2} \) |
| 17 | \( 1 - 17.8T + 289T^{2} \) |
| 19 | \( 1 - 5.19iT - 361T^{2} \) |
| 23 | \( 1 + 24.9iT - 529T^{2} \) |
| 29 | \( 1 + 29.6T + 841T^{2} \) |
| 31 | \( 1 + 17.1iT - 961T^{2} \) |
| 37 | \( 1 + 6.41T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.64T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 52.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.41T + 3.72e3T^{2} \) |
| 67 | \( 1 + 29.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 2.16T + 5.32e3T^{2} \) |
| 79 | \( 1 - 149. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−9.049353007330859758524282008526, −8.428692036150714682809316006621, −8.020224023505787327991812831827, −7.02880372376445964292080708299, −5.96898339189472906850239970548, −5.41926861509290333292488869287, −4.14240586013002985746254707226, −3.50045326656312215665885232592, −2.52124337167799671472798397534, −0.951801400777373354554397282505,
0.42275805772104576452421435694, 1.48231996053303544900358696692, 3.38014822770323754412145928936, 3.77028246214926819643069136122, 4.55810555473958332153694919209, 5.64001145803046136627334957716, 6.86996278518172150205663284744, 7.46323576530693101788199058226, 7.892218852601748555863874059987, 8.827754018402138923903773726502
