| L(s) = 1 | − 729·3-s − 1.56e4·5-s − 2.76e5·7-s + 5.31e5·9-s − 1.25e6·11-s − 1.87e7·13-s + 1.13e7·15-s − 5.70e7·17-s − 2.27e8·19-s + 2.01e8·21-s + 2.39e8·23-s + 2.44e8·25-s − 3.87e8·27-s − 1.95e9·29-s − 5.88e9·31-s + 9.16e8·33-s + 4.32e9·35-s − 4.98e8·37-s + 1.36e10·39-s − 2.29e10·41-s + 1.56e10·43-s − 8.30e9·45-s + 8.59e10·47-s − 2.03e10·49-s + 4.16e10·51-s − 1.74e10·53-s + 1.96e10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.888·7-s + 1/3·9-s − 0.213·11-s − 1.07·13-s + 0.258·15-s − 0.573·17-s − 1.11·19-s + 0.513·21-s + 0.337·23-s + 1/5·25-s − 0.192·27-s − 0.609·29-s − 1.19·31-s + 0.123·33-s + 0.397·35-s − 0.0319·37-s + 0.620·39-s − 0.754·41-s + 0.378·43-s − 0.149·45-s + 1.16·47-s − 0.209·49-s + 0.331·51-s − 0.108·53-s + 0.0956·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.5481768952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5481768952\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{6} T \) |
| 5 | \( 1 + p^{6} T \) |
| good | 7 | \( 1 + 276676 T + p^{13} T^{2} \) |
| 11 | \( 1 + 114264 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 18708910 T + p^{13} T^{2} \) |
| 17 | \( 1 + 57090426 T + p^{13} T^{2} \) |
| 19 | \( 1 + 227676100 T + p^{13} T^{2} \) |
| 23 | \( 1 - 239871624 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1953921846 T + p^{13} T^{2} \) |
| 31 | \( 1 + 5882983096 T + p^{13} T^{2} \) |
| 37 | \( 1 + 498430798 T + p^{13} T^{2} \) |
| 41 | \( 1 + 22962871110 T + p^{13} T^{2} \) |
| 43 | \( 1 - 15678111284 T + p^{13} T^{2} \) |
| 47 | \( 1 - 85979092296 T + p^{13} T^{2} \) |
| 53 | \( 1 + 17457596502 T + p^{13} T^{2} \) |
| 59 | \( 1 - 3227146824 T + p^{13} T^{2} \) |
| 61 | \( 1 - 528201214022 T + p^{13} T^{2} \) |
| 67 | \( 1 - 1124989657076 T + p^{13} T^{2} \) |
| 71 | \( 1 + 628595885280 T + p^{13} T^{2} \) |
| 73 | \( 1 - 325101409922 T + p^{13} T^{2} \) |
| 79 | \( 1 + 805411437832 T + p^{13} T^{2} \) |
| 83 | \( 1 - 3982895195604 T + p^{13} T^{2} \) |
| 89 | \( 1 - 8178637478610 T + p^{13} T^{2} \) |
| 97 | \( 1 + 1875170155102 T + p^{13} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43052081925969213020363404578, −11.22334741185705914656246456924, −10.17490651902902924085330539301, −8.992643992808189468541821328786, −7.44367283568845577525184150112, −6.45063380597434725309300924651, −5.08068979748163231663111829900, −3.78091497364902857261959964043, −2.26381204100198479630202335834, −0.38270925400258442331882633770,
0.38270925400258442331882633770, 2.26381204100198479630202335834, 3.78091497364902857261959964043, 5.08068979748163231663111829900, 6.45063380597434725309300924651, 7.44367283568845577525184150112, 8.992643992808189468541821328786, 10.17490651902902924085330539301, 11.22334741185705914656246456924, 12.43052081925969213020363404578
