| L(s) = 1 | + 0.785·3-s + 1.28·5-s − 3.55·7-s − 2.38·9-s + 11-s − 1.72·13-s + 1.00·15-s + 3.23·17-s − 0.00937·19-s − 2.79·21-s − 7.62·23-s − 3.35·25-s − 4.22·27-s + 2.21·29-s + 5.55·31-s + 0.785·33-s − 4.55·35-s − 0.632·37-s − 1.35·39-s + 8.90·41-s + 10.4·43-s − 3.05·45-s + 0.474·47-s + 5.63·49-s + 2.54·51-s + 3.85·53-s + 1.28·55-s + ⋯ |
| L(s) = 1 | + 0.453·3-s + 0.573·5-s − 1.34·7-s − 0.794·9-s + 0.301·11-s − 0.478·13-s + 0.260·15-s + 0.785·17-s − 0.00215·19-s − 0.609·21-s − 1.59·23-s − 0.671·25-s − 0.813·27-s + 0.410·29-s + 0.997·31-s + 0.136·33-s − 0.770·35-s − 0.104·37-s − 0.217·39-s + 1.39·41-s + 1.59·43-s − 0.455·45-s + 0.0692·47-s + 0.805·49-s + 0.356·51-s + 0.529·53-s + 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.726073063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726073063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 3 | \( 1 - 0.785T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 13 | \( 1 + 1.72T + 13T^{2} \) |
| 17 | \( 1 - 3.23T + 17T^{2} \) |
| 19 | \( 1 + 0.00937T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 + 0.632T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 0.474T + 47T^{2} \) |
| 53 | \( 1 - 3.85T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 5.28T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 - 4.81T + 83T^{2} \) |
| 89 | \( 1 - 5.18T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.997742999383797744152796030684, −7.51780749360908910047466960633, −6.42764753604444897405834511731, −6.04923419012402092545046561659, −5.45602286750331569947830473272, −4.24837411880136439458582284660, −3.53877476951710668013845039483, −2.73385164058257394309787544323, −2.13341322862717072185406547275, −0.64337202599549176904945976705,
0.64337202599549176904945976705, 2.13341322862717072185406547275, 2.73385164058257394309787544323, 3.53877476951710668013845039483, 4.24837411880136439458582284660, 5.45602286750331569947830473272, 6.04923419012402092545046561659, 6.42764753604444897405834511731, 7.51780749360908910047466960633, 7.997742999383797744152796030684
