| L(s) = 1 | − 0.596·2-s − 3-s − 1.64·4-s + 4.06·5-s + 0.596·6-s − 7-s + 2.17·8-s + 9-s − 2.42·10-s − 0.0366·11-s + 1.64·12-s + 5.73·13-s + 0.596·14-s − 4.06·15-s + 1.99·16-s + 3.28·17-s − 0.596·18-s − 0.844·19-s − 6.67·20-s + 21-s + 0.0218·22-s − 2.13·23-s − 2.17·24-s + 11.4·25-s − 3.42·26-s − 27-s + 1.64·28-s + ⋯ |
| L(s) = 1 | − 0.421·2-s − 0.577·3-s − 0.822·4-s + 1.81·5-s + 0.243·6-s − 0.377·7-s + 0.768·8-s + 0.333·9-s − 0.766·10-s − 0.0110·11-s + 0.474·12-s + 1.59·13-s + 0.159·14-s − 1.04·15-s + 0.497·16-s + 0.797·17-s − 0.140·18-s − 0.193·19-s − 1.49·20-s + 0.218·21-s + 0.00466·22-s − 0.445·23-s − 0.443·24-s + 2.29·25-s − 0.671·26-s − 0.192·27-s + 0.310·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674107410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674107410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 + 0.596T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 11 | \( 1 + 0.0366T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 0.844T + 19T^{2} \) |
| 23 | \( 1 + 2.13T + 23T^{2} \) |
| 29 | \( 1 - 7.14T + 29T^{2} \) |
| 31 | \( 1 - 6.50T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 4.01T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 5.39T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 5.56T + 73T^{2} \) |
| 79 | \( 1 - 3.28T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.798816817106711525673869373892, −7.86120426262609097777927635660, −6.73710052812403558714813457227, −6.10274295121068227361150781841, −5.64991537839988289867267658365, −4.87384830894960300823819314615, −3.94388077519321593107566449680, −2.86132160010638896070436919000, −1.58103156763511513472216267155, −0.917638996280111379297518202651,
0.917638996280111379297518202651, 1.58103156763511513472216267155, 2.86132160010638896070436919000, 3.94388077519321593107566449680, 4.87384830894960300823819314615, 5.64991537839988289867267658365, 6.10274295121068227361150781841, 6.73710052812403558714813457227, 7.86120426262609097777927635660, 8.798816817106711525673869373892
