| L(s) = 1 | − 9·3-s − 25·5-s − 160·7-s + 81·9-s + 596·11-s + 122·13-s + 225·15-s − 1.07e3·17-s − 796·19-s + 1.44e3·21-s − 1.08e3·23-s + 625·25-s − 729·27-s − 46·29-s − 4.95e3·31-s − 5.36e3·33-s + 4.00e3·35-s + 6.11e3·37-s − 1.09e3·39-s − 6·41-s + 2.41e4·43-s − 2.02e3·45-s + 1.34e4·47-s + 8.79e3·49-s + 9.70e3·51-s − 2.05e4·53-s − 1.49e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.23·7-s + 1/3·9-s + 1.48·11-s + 0.200·13-s + 0.258·15-s − 0.904·17-s − 0.505·19-s + 0.712·21-s − 0.428·23-s + 1/5·25-s − 0.192·27-s − 0.0101·29-s − 0.925·31-s − 0.857·33-s + 0.551·35-s + 0.734·37-s − 0.115·39-s − 0.000557·41-s + 1.98·43-s − 0.149·45-s + 0.890·47-s + 0.523·49-s + 0.522·51-s − 1.00·53-s − 0.664·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 160 T + p^{5} T^{2} \) |
| 11 | \( 1 - 596 T + p^{5} T^{2} \) |
| 13 | \( 1 - 122 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1078 T + p^{5} T^{2} \) |
| 19 | \( 1 + 796 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1088 T + p^{5} T^{2} \) |
| 29 | \( 1 + 46 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4952 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6114 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6 T + p^{5} T^{2} \) |
| 43 | \( 1 - 24116 T + p^{5} T^{2} \) |
| 47 | \( 1 - 13480 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20598 T + p^{5} T^{2} \) |
| 59 | \( 1 - 46756 T + p^{5} T^{2} \) |
| 61 | \( 1 - 9602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17404 T + p^{5} T^{2} \) |
| 71 | \( 1 - 26568 T + p^{5} T^{2} \) |
| 73 | \( 1 - 75450 T + p^{5} T^{2} \) |
| 79 | \( 1 - 50472 T + p^{5} T^{2} \) |
| 83 | \( 1 + 33236 T + p^{5} T^{2} \) |
| 89 | \( 1 - 133194 T + p^{5} T^{2} \) |
| 97 | \( 1 + 42878 T + p^{5} T^{2} \) |
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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
−9.183088291431652236219991727630, −8.005872877382390355006183258460, −6.78764991306893211687995567098, −6.53401247348301378154699913876, −5.56188297592693380905721522498, −4.16382195959681808049909710140, −3.77216660241637920842221483366, −2.37728846958309239930954680383, −0.984156761247429773510903205610, 0,
0.984156761247429773510903205610, 2.37728846958309239930954680383, 3.77216660241637920842221483366, 4.16382195959681808049909710140, 5.56188297592693380905721522498, 6.53401247348301378154699913876, 6.78764991306893211687995567098, 8.005872877382390355006183258460, 9.183088291431652236219991727630
