| L(s) = 1 | − 0.213·2-s + 3·3-s − 7.95·4-s − 15.3·5-s − 0.641·6-s + 32.3·7-s + 3.40·8-s + 9·9-s + 3.27·10-s − 29.5·11-s − 23.8·12-s − 6.92·14-s − 46.0·15-s + 62.9·16-s − 78.1·17-s − 1.92·18-s − 10.6·19-s + 122.·20-s + 97.1·21-s + 6.32·22-s − 26.8·23-s + 10.2·24-s + 110.·25-s + 27·27-s − 257.·28-s − 190.·29-s + 9.83·30-s + ⋯ |
| L(s) = 1 | − 0.0755·2-s + 0.577·3-s − 0.994·4-s − 1.37·5-s − 0.0436·6-s + 1.74·7-s + 0.150·8-s + 0.333·9-s + 0.103·10-s − 0.811·11-s − 0.574·12-s − 0.132·14-s − 0.792·15-s + 0.982·16-s − 1.11·17-s − 0.0251·18-s − 0.128·19-s + 1.36·20-s + 1.00·21-s + 0.0612·22-s − 0.243·23-s + 0.0869·24-s + 0.882·25-s + 0.192·27-s − 1.73·28-s − 1.22·29-s + 0.0598·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.445978753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445978753\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.213T + 8T^{2} \) |
| 5 | \( 1 + 15.3T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 11 | \( 1 + 29.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 78.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 26.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 464.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 248.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 740.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 340.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 590.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 340.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 36.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 164.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 327.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.40e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 736.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−10.70395818906654728654116209185, −9.329497961112365460997579212905, −8.513372719958250644451414402113, −7.893246024464252676168383620264, −7.47237691119821399684602852046, −5.51982030091184697726478454171, −4.35136094317743944417155972579, −4.15119282799731212503161629197, −2.41534315070755980712694849592, −0.74532184622019125158232130820,
0.74532184622019125158232130820, 2.41534315070755980712694849592, 4.15119282799731212503161629197, 4.35136094317743944417155972579, 5.51982030091184697726478454171, 7.47237691119821399684602852046, 7.893246024464252676168383620264, 8.513372719958250644451414402113, 9.329497961112365460997579212905, 10.70395818906654728654116209185
