| L(s) = 1 | − 2.47·3-s − 5-s − 7-s + 3.11·9-s + 3.58·11-s + 13-s + 2.47·15-s − 7.70·17-s − 1.47·19-s + 2.47·21-s − 2.87·23-s + 25-s − 0.284·27-s + 8.29·29-s + 6.47·31-s − 8.87·33-s + 35-s − 7.75·37-s − 2.47·39-s − 4.34·41-s + 0.945·43-s − 3.11·45-s + 8.15·47-s + 49-s + 19.0·51-s + 5.17·53-s − 3.58·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 0.447·5-s − 0.377·7-s + 1.03·9-s + 1.08·11-s + 0.277·13-s + 0.638·15-s − 1.86·17-s − 0.337·19-s + 0.539·21-s − 0.598·23-s + 0.200·25-s − 0.0546·27-s + 1.53·29-s + 1.16·31-s − 1.54·33-s + 0.169·35-s − 1.27·37-s − 0.395·39-s − 0.678·41-s + 0.144·43-s − 0.464·45-s + 1.18·47-s + 0.142·49-s + 2.66·51-s + 0.710·53-s − 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 - 0.945T + 43T^{2} \) |
| 47 | \( 1 - 8.15T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 1.45T + 59T^{2} \) |
| 61 | \( 1 + 9.58T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.06T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 1.60T + 89T^{2} \) |
| 97 | \( 1 + 9.24T + 97T^{2} \) |
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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
−8.314053787790308873324009933418, −7.01631279828185222672799976139, −6.55122751257959955769606582167, −6.19012579089818219402307887557, −5.09976910111768756783504393044, −4.41678277950667169391252262666, −3.76539612772925655093572598668, −2.43105821699929996494480473551, −1.09116086009730539222697759362, 0,
1.09116086009730539222697759362, 2.43105821699929996494480473551, 3.76539612772925655093572598668, 4.41678277950667169391252262666, 5.09976910111768756783504393044, 6.19012579089818219402307887557, 6.55122751257959955769606582167, 7.01631279828185222672799976139, 8.314053787790308873324009933418
