| L(s) = 1 | − 2.50·2-s + 2.09·3-s + 4.28·4-s − 3.15·5-s − 5.24·6-s − 2.77·7-s − 5.74·8-s + 1.37·9-s + 7.91·10-s + 2.37·11-s + 8.97·12-s + 1.38·13-s + 6.95·14-s − 6.60·15-s + 5.81·16-s − 2.15·17-s − 3.45·18-s − 2.24·19-s − 13.5·20-s − 5.80·21-s − 5.94·22-s + 5.80·23-s − 12.0·24-s + 4.96·25-s − 3.48·26-s − 3.39·27-s − 11.8·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 1.20·3-s + 2.14·4-s − 1.41·5-s − 2.14·6-s − 1.04·7-s − 2.02·8-s + 0.458·9-s + 2.50·10-s + 0.715·11-s + 2.59·12-s + 0.385·13-s + 1.85·14-s − 1.70·15-s + 1.45·16-s − 0.523·17-s − 0.813·18-s − 0.515·19-s − 3.02·20-s − 1.26·21-s − 1.26·22-s + 1.21·23-s − 2.45·24-s + 0.993·25-s − 0.683·26-s − 0.653·27-s − 2.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5558138436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5558138436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 5.80T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 0.699T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 + 8.25T + 59T^{2} \) |
| 61 | \( 1 + 0.889T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 + 8.37T + 97T^{2} \) |
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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
−8.714462321560813317971926826850, −7.86508957018141353561331347628, −7.26264751968129322386463352993, −6.88372894274526921392943802177, −5.87594029362279500490168735145, −4.18600349880051604988909627972, −3.52444353445061691131010401908, −2.84556690835970973753115297953, −1.82190886737407865607816436243, −0.50817971577358205763994859347,
0.50817971577358205763994859347, 1.82190886737407865607816436243, 2.84556690835970973753115297953, 3.52444353445061691131010401908, 4.18600349880051604988909627972, 5.87594029362279500490168735145, 6.88372894274526921392943802177, 7.26264751968129322386463352993, 7.86508957018141353561331347628, 8.714462321560813317971926826850
