| L(s) = 1 | − 2.16·2-s + 3·3-s − 3.32·4-s + 7.38·5-s − 6.48·6-s − 25.4·7-s + 24.4·8-s + 9·9-s − 15.9·10-s − 11·11-s − 9.98·12-s − 88.8·13-s + 55.0·14-s + 22.1·15-s − 26.2·16-s − 27.5·17-s − 19.4·18-s − 90.6·19-s − 24.5·20-s − 76.3·21-s + 23.7·22-s + 5.00·23-s + 73.4·24-s − 70.4·25-s + 192.·26-s + 27·27-s + 84.7·28-s + ⋯ |
| L(s) = 1 | − 0.764·2-s + 0.577·3-s − 0.416·4-s + 0.660·5-s − 0.441·6-s − 1.37·7-s + 1.08·8-s + 0.333·9-s − 0.504·10-s − 0.301·11-s − 0.240·12-s − 1.89·13-s + 1.05·14-s + 0.381·15-s − 0.410·16-s − 0.393·17-s − 0.254·18-s − 1.09·19-s − 0.274·20-s − 0.793·21-s + 0.230·22-s + 0.0454·23-s + 0.624·24-s − 0.563·25-s + 1.44·26-s + 0.192·27-s + 0.571·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4362458239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4362458239\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 2.16T + 8T^{2} \) |
| 5 | \( 1 - 7.38T + 125T^{2} \) |
| 7 | \( 1 + 25.4T + 343T^{2} \) |
| 13 | \( 1 + 88.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 5.00T + 1.21e4T^{2} \) |
| 29 | \( 1 + 298.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 349.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 549.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 625.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 244.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 36.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 384.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 880.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 955.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 266.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 86.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−9.111010960037767852976520014431, −8.091434930003709037897970287811, −7.39565521246756783410648368953, −6.65915614118959371977928920468, −5.66273298084152084199422716068, −4.68384324411551333750877628747, −3.79784369224719311045376645783, −2.61190069744333862407106593494, −1.93615935649450117584772605142, −0.31315686806178016617257021672,
0.31315686806178016617257021672, 1.93615935649450117584772605142, 2.61190069744333862407106593494, 3.79784369224719311045376645783, 4.68384324411551333750877628747, 5.66273298084152084199422716068, 6.65915614118959371977928920468, 7.39565521246756783410648368953, 8.091434930003709037897970287811, 9.111010960037767852976520014431
