| L(s) = 1 | − 29.1·2-s − 243·3-s − 1.20e3·4-s + 7.07e3·6-s + 5.66e4·7-s + 9.45e4·8-s + 5.90e4·9-s + 2.35e5·11-s + 2.91e5·12-s + 1.14e6·13-s − 1.64e6·14-s − 2.94e5·16-s + 1.19e6·17-s − 1.71e6·18-s − 1.61e7·19-s − 1.37e7·21-s − 6.85e6·22-s − 9.03e6·23-s − 2.29e7·24-s − 3.32e7·26-s − 1.43e7·27-s − 6.79e7·28-s + 8.87e7·29-s + 1.93e8·31-s − 1.85e8·32-s − 5.72e7·33-s − 3.47e7·34-s + ⋯ |
| L(s) = 1 | − 0.643·2-s − 0.577·3-s − 0.586·4-s + 0.371·6-s + 1.27·7-s + 1.02·8-s + 0.333·9-s + 0.441·11-s + 0.338·12-s + 0.854·13-s − 0.819·14-s − 0.0702·16-s + 0.203·17-s − 0.214·18-s − 1.49·19-s − 0.735·21-s − 0.283·22-s − 0.292·23-s − 0.589·24-s − 0.549·26-s − 0.192·27-s − 0.746·28-s + 0.803·29-s + 1.21·31-s − 0.975·32-s − 0.254·33-s − 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.214467503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214467503\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 29.1T + 2.04e3T^{2} \) |
| 7 | \( 1 - 5.66e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.35e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.14e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.19e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.61e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 9.03e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 8.87e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.93e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.44e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.12e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.52e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.98e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 9.38e7T + 9.26e18T^{2} \) |
| 59 | \( 1 + 8.60e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 7.84e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 9.98e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.24e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.51e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.32e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.08e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.35e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.52e10T + 7.15e21T^{2} \) |
| 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
−12.03960885655089782133118308341, −10.98454099225972669078193178882, −10.13798796799360722236226417122, −8.691016716205048242175665642646, −8.048536889804648666257920605524, −6.47922022096381992570981758548, −5.00852197931380953539780603832, −4.08619792739704724199564706431, −1.73856443960233073786564603551, −0.73326570676871399340409692730,
0.73326570676871399340409692730, 1.73856443960233073786564603551, 4.08619792739704724199564706431, 5.00852197931380953539780603832, 6.47922022096381992570981758548, 8.048536889804648666257920605524, 8.691016716205048242175665642646, 10.13798796799360722236226417122, 10.98454099225972669078193178882, 12.03960885655089782133118308341
