| L(s) = 1 | + 9.65·2-s − 22.8·3-s + 61.1·4-s − 25·5-s − 220.·6-s + 281.·8-s + 279.·9-s − 241.·10-s + 146.·11-s − 1.39e3·12-s − 761.·13-s + 571.·15-s + 759.·16-s + 2.32e3·17-s + 2.69e3·18-s + 1.71e3·19-s − 1.52e3·20-s + 1.41e3·22-s − 270.·23-s − 6.43e3·24-s + 625·25-s − 7.34e3·26-s − 835.·27-s + 6.71e3·29-s + 5.51e3·30-s − 1.38e3·31-s − 1.67e3·32-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 1.46·3-s + 1.91·4-s − 0.447·5-s − 2.50·6-s + 1.55·8-s + 1.15·9-s − 0.763·10-s + 0.364·11-s − 2.80·12-s − 1.24·13-s + 0.655·15-s + 0.741·16-s + 1.95·17-s + 1.96·18-s + 1.09·19-s − 0.854·20-s + 0.621·22-s − 0.106·23-s − 2.27·24-s + 0.200·25-s − 2.13·26-s − 0.220·27-s + 1.48·29-s + 1.11·30-s − 0.259·31-s − 0.289·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.222609664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222609664\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 9.65T + 32T^{2} \) |
| 3 | \( 1 + 22.8T + 243T^{2} \) |
| 11 | \( 1 - 146.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 761.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.32e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 270.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.38e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.64e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.19e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.08e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.29e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.51e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.81e4T + 8.58e9T^{2} \) |
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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
−11.80710991857152848809107563095, −10.77467982207349943715024484991, −9.757045970128028698275091488511, −7.64771844262168564057348572251, −6.81005867135896251151763721260, −5.69359150792954874632323374135, −5.16587719621954017148730579563, −4.15698405654729641985932855083, −2.89073418380423223982098965376, −0.905583554050149278598374000749,
0.905583554050149278598374000749, 2.89073418380423223982098965376, 4.15698405654729641985932855083, 5.16587719621954017148730579563, 5.69359150792954874632323374135, 6.81005867135896251151763721260, 7.64771844262168564057348572251, 9.757045970128028698275091488511, 10.77467982207349943715024484991, 11.80710991857152848809107563095
