| L(s) = 1 | − 0.411·2-s − 1.83·4-s + 3.83·5-s − 3.43·7-s + 1.57·8-s − 1.57·10-s − 4.43·11-s + 2.14·13-s + 1.41·14-s + 3.01·16-s + 4.41·17-s − 6.36·19-s − 7.01·20-s + 1.82·22-s − 2.92·23-s + 9.69·25-s − 0.882·26-s + 6.29·28-s + 2.96·29-s + 5.41·31-s − 4.39·32-s − 1.82·34-s − 13.1·35-s + 6.78·37-s + 2.62·38-s + 6.04·40-s − 2.91·41-s + ⋯ |
| L(s) = 1 | − 0.291·2-s − 0.915·4-s + 1.71·5-s − 1.29·7-s + 0.557·8-s − 0.499·10-s − 1.33·11-s + 0.593·13-s + 0.378·14-s + 0.752·16-s + 1.07·17-s − 1.46·19-s − 1.56·20-s + 0.389·22-s − 0.610·23-s + 1.93·25-s − 0.173·26-s + 1.18·28-s + 0.551·29-s + 0.971·31-s − 0.777·32-s − 0.312·34-s − 2.22·35-s + 1.11·37-s + 0.425·38-s + 0.956·40-s − 0.454·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.288748686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.288748686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.411T + 2T^{2} \) |
| 5 | \( 1 - 3.83T + 5T^{2} \) |
| 7 | \( 1 + 3.43T + 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 - 4.41T + 17T^{2} \) |
| 19 | \( 1 + 6.36T + 19T^{2} \) |
| 23 | \( 1 + 2.92T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 - 9.64T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 5.42T + 67T^{2} \) |
| 71 | \( 1 - 7.67T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 - 3.30T + 79T^{2} \) |
| 83 | \( 1 + 9.22T + 83T^{2} \) |
| 89 | \( 1 + 0.615T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.232452738637964872282704114165, −8.488958326128343540424592694192, −7.70741812200770976440631318914, −6.43164201914423965708127230723, −5.97031223414519519974471907850, −5.28813911249133025804913908084, −4.26061443235000266350291258442, −3.07866763656137297360458723711, −2.22122655795198965792880100700, −0.77782400340100948676482026473,
0.77782400340100948676482026473, 2.22122655795198965792880100700, 3.07866763656137297360458723711, 4.26061443235000266350291258442, 5.28813911249133025804913908084, 5.97031223414519519974471907850, 6.43164201914423965708127230723, 7.70741812200770976440631318914, 8.488958326128343540424592694192, 9.232452738637964872282704114165
