| L(s) = 1 | − 0.278·3-s − 5-s − 2.48·7-s − 2.92·9-s − 4.96·11-s + 3.11·13-s + 0.278·15-s + 5.66·17-s + 6.08·19-s + 0.692·21-s + 1.52·23-s + 25-s + 1.64·27-s − 3.59·29-s + 2.97·31-s + 1.38·33-s + 2.48·35-s − 2.11·37-s − 0.866·39-s − 7.11·41-s + 5.15·43-s + 2.92·45-s + 11.0·47-s − 0.807·49-s − 1.57·51-s − 4.27·53-s + 4.96·55-s + ⋯ |
| L(s) = 1 | − 0.160·3-s − 0.447·5-s − 0.940·7-s − 0.974·9-s − 1.49·11-s + 0.863·13-s + 0.0718·15-s + 1.37·17-s + 1.39·19-s + 0.151·21-s + 0.318·23-s + 0.200·25-s + 0.317·27-s − 0.667·29-s + 0.534·31-s + 0.240·33-s + 0.420·35-s − 0.347·37-s − 0.138·39-s − 1.11·41-s + 0.785·43-s + 0.435·45-s + 1.61·47-s − 0.115·49-s − 0.220·51-s − 0.586·53-s + 0.669·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 0.278T + 3T^{2} \) |
| 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 7.11T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 4.27T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 + 2.41T + 61T^{2} \) |
| 67 | \( 1 - 5.86T + 67T^{2} \) |
| 71 | \( 1 - 5.61T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 + 9.74T + 79T^{2} \) |
| 83 | \( 1 - 8.75T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 + 4.79T + 97T^{2} \) |
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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
−7.76416114094323002769388408756, −7.12204151803851482420774545912, −6.16951701168867255782421641529, −5.51095297045949080530659659193, −5.13726470553699643801005145477, −3.79860911018434862832749776945, −3.17503894206049791187909414784, −2.66337540215459799120401282089, −1.07780819104178926137905141863, 0,
1.07780819104178926137905141863, 2.66337540215459799120401282089, 3.17503894206049791187909414784, 3.79860911018434862832749776945, 5.13726470553699643801005145477, 5.51095297045949080530659659193, 6.16951701168867255782421641529, 7.12204151803851482420774545912, 7.76416114094323002769388408756
