| L(s) = 1 | + 0.813·3-s − 3.10·5-s + 3.10·7-s − 2.33·9-s − 5.10·11-s + 0.289·13-s − 2.52·15-s − 1.10·17-s − 7.04·19-s + 2.52·21-s − 23-s + 4.62·25-s − 4.34·27-s + 7.54·29-s + 0.235·31-s − 4.15·33-s − 9.62·35-s − 1.94·37-s + 0.235·39-s − 4.28·41-s − 1.42·43-s + 7.25·45-s − 11.0·47-s + 2.62·49-s − 0.897·51-s − 11.8·53-s + 15.8·55-s + ⋯ |
| L(s) = 1 | + 0.469·3-s − 1.38·5-s + 1.17·7-s − 0.779·9-s − 1.53·11-s + 0.0802·13-s − 0.651·15-s − 0.267·17-s − 1.61·19-s + 0.550·21-s − 0.208·23-s + 0.925·25-s − 0.835·27-s + 1.40·29-s + 0.0422·31-s − 0.722·33-s − 1.62·35-s − 0.319·37-s + 0.0376·39-s − 0.669·41-s − 0.216·43-s + 1.08·45-s − 1.60·47-s + 0.375·49-s − 0.125·51-s − 1.62·53-s + 2.13·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.813T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 - 0.289T + 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 + 7.04T + 19T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 0.235T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 6.67T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 4.81T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 + 8.88T + 79T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 4.05T + 97T^{2} \) |
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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
−10.10593897158554552487498459174, −8.540030518351843601829783104309, −8.269089680963031772637643940311, −7.78038841979252055262804463547, −6.55346140188520437677880242418, −5.15637539375186187163300803549, −4.47033339906226613982322669618, −3.29484937655835717959378666532, −2.19169933003470649118460555906, 0,
2.19169933003470649118460555906, 3.29484937655835717959378666532, 4.47033339906226613982322669618, 5.15637539375186187163300803549, 6.55346140188520437677880242418, 7.78038841979252055262804463547, 8.269089680963031772637643940311, 8.540030518351843601829783104309, 10.10593897158554552487498459174
