| L(s) = 1 | − 3-s + 3.61·5-s + 2.72·7-s + 9-s + 4.28·11-s − 0.0474·13-s − 3.61·15-s + 3.70·17-s + 0.502·19-s − 2.72·21-s + 2.97·23-s + 8.08·25-s − 27-s − 6.25·29-s + 5.93·31-s − 4.28·33-s + 9.86·35-s + 0.158·37-s + 0.0474·39-s − 3.89·41-s + 4.94·43-s + 3.61·45-s + 8.39·47-s + 0.440·49-s − 3.70·51-s − 10.7·53-s + 15.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.61·5-s + 1.03·7-s + 0.333·9-s + 1.29·11-s − 0.0131·13-s − 0.933·15-s + 0.899·17-s + 0.115·19-s − 0.595·21-s + 0.621·23-s + 1.61·25-s − 0.192·27-s − 1.16·29-s + 1.06·31-s − 0.745·33-s + 1.66·35-s + 0.0260·37-s + 0.00759·39-s − 0.608·41-s + 0.754·43-s + 0.539·45-s + 1.22·47-s + 0.0629·49-s − 0.519·51-s − 1.47·53-s + 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.965032194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965032194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 + 0.0474T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 0.502T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 - 5.93T + 31T^{2} \) |
| 37 | \( 1 - 0.158T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 8.39T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 - 3.62T + 61T^{2} \) |
| 67 | \( 1 - 0.477T + 67T^{2} \) |
| 71 | \( 1 - 0.963T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 1.53T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 1.01T + 89T^{2} \) |
| 97 | \( 1 + 1.19T + 97T^{2} \) |
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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
−8.579888455971218181562514725374, −7.62024326498103754851205090350, −6.82319103833462300770220121127, −6.10489350366931817574730410096, −5.54832380751917946159145193788, −4.89451942312129584560563381538, −4.01786865269168335956579232528, −2.78568321783171688783345902309, −1.64037572365839893412781602268, −1.21087594898677244403285284082,
1.21087594898677244403285284082, 1.64037572365839893412781602268, 2.78568321783171688783345902309, 4.01786865269168335956579232528, 4.89451942312129584560563381538, 5.54832380751917946159145193788, 6.10489350366931817574730410096, 6.82319103833462300770220121127, 7.62024326498103754851205090350, 8.579888455971218181562514725374
