| L(s) = 1 | − 2.11·3-s + 1.77·5-s − 7-s + 1.45·9-s + 0.612·11-s + 1.84·13-s − 3.75·15-s + 4.26·17-s − 4.70·19-s + 2.11·21-s − 23-s − 1.83·25-s + 3.25·27-s + 10.2·29-s + 10.2·31-s − 1.29·33-s − 1.77·35-s − 2.47·37-s − 3.90·39-s + 6.37·41-s + 10.9·43-s + 2.59·45-s + 0.544·47-s + 49-s − 9.00·51-s − 7.61·53-s + 1.09·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 0.795·5-s − 0.377·7-s + 0.486·9-s + 0.184·11-s + 0.512·13-s − 0.969·15-s + 1.03·17-s − 1.07·19-s + 0.460·21-s − 0.208·23-s − 0.367·25-s + 0.625·27-s + 1.90·29-s + 1.83·31-s − 0.225·33-s − 0.300·35-s − 0.407·37-s − 0.624·39-s + 0.995·41-s + 1.66·43-s + 0.387·45-s + 0.0794·47-s + 0.142·49-s − 1.26·51-s − 1.04·53-s + 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093314590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093314590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 11 | \( 1 - 0.612T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 6.37T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 0.544T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 5.27T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 - 5.84T + 83T^{2} \) |
| 89 | \( 1 - 0.00746T + 89T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−10.42887319299569093144908188912, −10.05424098071921848026707816498, −8.934078314862118381011105495846, −7.927977123033342820723553035922, −6.44392132234959473058426412852, −6.23627460997414417135387966788, −5.28430117269705697168936253973, −4.21252806272591130336908690675, −2.67281038292489146725014239221, −0.993744029390338225978274947159,
0.993744029390338225978274947159, 2.67281038292489146725014239221, 4.21252806272591130336908690675, 5.28430117269705697168936253973, 6.23627460997414417135387966788, 6.44392132234959473058426412852, 7.927977123033342820723553035922, 8.934078314862118381011105495846, 10.05424098071921848026707816498, 10.42887319299569093144908188912
