| L(s) = 1 | + 4.79·2-s − 3·3-s + 14.9·4-s − 13.3·5-s − 14.3·6-s − 17.5·7-s + 33.4·8-s + 9·9-s − 63.8·10-s − 11·11-s − 44.9·12-s + 6.48·13-s − 84.0·14-s + 39.9·15-s + 40.4·16-s + 41.2·17-s + 43.1·18-s − 70.7·19-s − 199.·20-s + 52.6·21-s − 52.7·22-s + 108.·23-s − 100.·24-s + 52.1·25-s + 31.0·26-s − 27·27-s − 262.·28-s + ⋯ |
| L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.87·4-s − 1.19·5-s − 0.978·6-s − 0.947·7-s + 1.47·8-s + 0.333·9-s − 2.01·10-s − 0.301·11-s − 1.08·12-s + 0.138·13-s − 1.60·14-s + 0.687·15-s + 0.632·16-s + 0.588·17-s + 0.564·18-s − 0.854·19-s − 2.22·20-s + 0.546·21-s − 0.510·22-s + 0.985·23-s − 0.853·24-s + 0.417·25-s + 0.234·26-s − 0.192·27-s − 1.77·28-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
| \(L(2)\) |
\(\approx\) |
\(2.747867110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.747867110\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.79T + 8T^{2} \) |
| 5 | \( 1 + 13.3T + 125T^{2} \) |
| 7 | \( 1 + 17.5T + 343T^{2} \) |
| 13 | \( 1 - 6.48T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 26.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 410.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.39T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 342.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 525.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 638.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 533.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 914.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 166.T + 9.12e5T^{2} \) |
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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
−8.669243601812485545375112338169, −7.58223240011833786243906340364, −6.95867243195434920692687131710, −6.26533547667392263252341263589, −5.45893937090216416319145367458, −4.72812766810717480754784811787, −3.79084895023544815747472188763, −3.42724527275133408307281704589, −2.29270124038516840897543241764, −0.57876780889365179422699818227,
0.57876780889365179422699818227, 2.29270124038516840897543241764, 3.42724527275133408307281704589, 3.79084895023544815747472188763, 4.72812766810717480754784811787, 5.45893937090216416319145367458, 6.26533547667392263252341263589, 6.95867243195434920692687131710, 7.58223240011833786243906340364, 8.669243601812485545375112338169
