| L(s) = 1 | − 3·5-s − 7-s − 3·11-s − 2·13-s − 3·17-s + 4·25-s − 4·31-s + 3·35-s − 5·37-s − 3·41-s + 10·43-s + 49-s + 12·53-s + 9·55-s − 6·59-s + 8·61-s + 6·65-s + 8·67-s + 14·73-s + 3·77-s − 4·79-s + 9·85-s − 3·89-s + 2·91-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.554·13-s − 0.727·17-s + 4/5·25-s − 0.718·31-s + 0.507·35-s − 0.821·37-s − 0.468·41-s + 1.52·43-s + 1/7·49-s + 1.64·53-s + 1.21·55-s − 0.781·59-s + 1.02·61-s + 0.744·65-s + 0.977·67-s + 1.63·73-s + 0.341·77-s − 0.450·79-s + 0.976·85-s − 0.317·89-s + 0.209·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7962882546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7962882546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.60736677374939, −12.07243493827867, −11.61089959519505, −11.16686444297275, −10.79339361900385, −10.30873620011751, −9.884396328391191, −9.222684071101642, −8.880803719918755, −8.298771136828069, −7.939536955736784, −7.481776085926666, −6.967552124329960, −6.789609155978428, −5.921628933155568, −5.462435184744293, −4.962430980517447, −4.459901484485434, −3.848842527383898, −3.625511467089743, −2.857430669904148, −2.426434988412993, −1.848232374589399, −0.8038057056173150, −0.3018133289581727,
0.3018133289581727, 0.8038057056173150, 1.848232374589399, 2.426434988412993, 2.857430669904148, 3.625511467089743, 3.848842527383898, 4.459901484485434, 4.962430980517447, 5.462435184744293, 5.921628933155568, 6.789609155978428, 6.967552124329960, 7.481776085926666, 7.939536955736784, 8.298771136828069, 8.880803719918755, 9.222684071101642, 9.884396328391191, 10.30873620011751, 10.79339361900385, 11.16686444297275, 11.61089959519505, 12.07243493827867, 12.60736677374939
