L(s) = 1 | + 1.59·3-s − 1.45·5-s − 2.10·7-s − 0.471·9-s − 0.978·11-s − 3.42·13-s − 2.30·15-s − 17-s − 7.73·19-s − 3.34·21-s − 2.83·23-s − 2.89·25-s − 5.51·27-s + 5.69·29-s − 0.490·31-s − 1.55·33-s + 3.05·35-s + 5.00·37-s − 5.45·39-s + 12.3·41-s − 3.39·43-s + 0.683·45-s + 8.86·47-s − 2.56·49-s − 1.59·51-s + 1.97·53-s + 1.41·55-s + ⋯ |
L(s) = 1 | + 0.918·3-s − 0.648·5-s − 0.795·7-s − 0.157·9-s − 0.294·11-s − 0.950·13-s − 0.595·15-s − 0.242·17-s − 1.77·19-s − 0.730·21-s − 0.591·23-s − 0.579·25-s − 1.06·27-s + 1.05·29-s − 0.0880·31-s − 0.270·33-s + 0.516·35-s + 0.822·37-s − 0.872·39-s + 1.92·41-s − 0.517·43-s + 0.101·45-s + 1.29·47-s − 0.367·49-s − 0.222·51-s + 0.270·53-s + 0.191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.107245292\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107245292\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 - 1.59T + 3T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 7 | \( 1 + 2.10T + 7T^{2} \) |
| 11 | \( 1 + 0.978T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 19 | \( 1 + 7.73T + 19T^{2} \) |
| 23 | \( 1 + 2.83T + 23T^{2} \) |
| 29 | \( 1 - 5.69T + 29T^{2} \) |
| 31 | \( 1 + 0.490T + 31T^{2} \) |
| 37 | \( 1 - 5.00T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 - 8.86T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 + 4.05T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 4.11T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.899710177432602542796689844354, −7.35982943756808959863432105811, −6.43242487081286131838386488331, −5.95549238425615976553419723360, −4.82615521873739669971667947346, −4.11449906655222063661738637546, −3.52593580395979677097109905708, −2.51541450350639394379739460119, −2.26171559328252475842502139086, −0.45816490317844076031368272714,
0.45816490317844076031368272714, 2.26171559328252475842502139086, 2.51541450350639394379739460119, 3.52593580395979677097109905708, 4.11449906655222063661738637546, 4.82615521873739669971667947346, 5.95549238425615976553419723360, 6.43242487081286131838386488331, 7.35982943756808959863432105811, 7.899710177432602542796689844354