| L(s) = 1 | − 3.08·2-s − 3·3-s + 1.52·4-s + 1.21·5-s + 9.25·6-s − 4.33·7-s + 19.9·8-s + 9·9-s − 3.73·10-s − 11·11-s − 4.56·12-s − 42.4·13-s + 13.3·14-s − 3.63·15-s − 73.8·16-s − 94.1·17-s − 27.7·18-s − 69.8·19-s + 1.84·20-s + 13.0·21-s + 33.9·22-s − 145.·23-s − 59.9·24-s − 123.·25-s + 130.·26-s − 27·27-s − 6.58·28-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 0.577·3-s + 0.190·4-s + 0.108·5-s + 0.629·6-s − 0.233·7-s + 0.883·8-s + 0.333·9-s − 0.118·10-s − 0.301·11-s − 0.109·12-s − 0.905·13-s + 0.255·14-s − 0.0625·15-s − 1.15·16-s − 1.34·17-s − 0.363·18-s − 0.843·19-s + 0.0205·20-s + 0.135·21-s + 0.328·22-s − 1.32·23-s − 0.510·24-s − 0.988·25-s + 0.987·26-s − 0.192·27-s − 0.0444·28-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
| \(L(2)\) |
\(\approx\) |
\(0.03508839770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03508839770\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.08T + 8T^{2} \) |
| 5 | \( 1 - 1.21T + 125T^{2} \) |
| 7 | \( 1 + 4.33T + 343T^{2} \) |
| 13 | \( 1 + 42.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 48.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 173.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 24.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 360.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 17.2T + 2.05e5T^{2} \) |
| 67 | \( 1 + 841.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 519.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 364.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 580.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 276.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 369.T + 9.12e5T^{2} \) |
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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
−8.790743229986991511012737744702, −8.163887949756816671305141441899, −7.31880107081167676049948771803, −6.62545254929310787906923870860, −5.74167161973557927996872481210, −4.67854136103045406259615601948, −4.09930666430607764794448300883, −2.46605530553318787258405150071, −1.63388054273432848111125407012, −0.10334516709402951669219295231,
0.10334516709402951669219295231, 1.63388054273432848111125407012, 2.46605530553318787258405150071, 4.09930666430607764794448300883, 4.67854136103045406259615601948, 5.74167161973557927996872481210, 6.62545254929310787906923870860, 7.31880107081167676049948771803, 8.163887949756816671305141441899, 8.790743229986991511012737744702
