| L(s) = 1 | − 3-s + 2.91·5-s + 0.844·7-s + 9-s − 2.29·11-s − 2.26·13-s − 2.91·15-s + 1.92·17-s − 7.01·19-s − 0.844·21-s + 2.00·23-s + 3.47·25-s − 27-s + 4.38·29-s − 4.17·31-s + 2.29·33-s + 2.45·35-s − 1.28·37-s + 2.26·39-s − 7.38·41-s + 1.02·43-s + 2.91·45-s + 8.09·47-s − 6.28·49-s − 1.92·51-s + 6.04·53-s − 6.69·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30·5-s + 0.319·7-s + 0.333·9-s − 0.693·11-s − 0.628·13-s − 0.751·15-s + 0.468·17-s − 1.60·19-s − 0.184·21-s + 0.417·23-s + 0.694·25-s − 0.192·27-s + 0.814·29-s − 0.750·31-s + 0.400·33-s + 0.415·35-s − 0.211·37-s + 0.363·39-s − 1.15·41-s + 0.156·43-s + 0.433·45-s + 1.18·47-s − 0.898·49-s − 0.270·51-s + 0.829·53-s − 0.902·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 151 | \( 1 + T \) |
| good | 5 | \( 1 - 2.91T + 5T^{2} \) |
| 7 | \( 1 - 0.844T + 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 1.28T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 - 1.02T + 43T^{2} \) |
| 47 | \( 1 - 8.09T + 47T^{2} \) |
| 53 | \( 1 - 6.04T + 53T^{2} \) |
| 59 | \( 1 - 0.803T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 + 15.0T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 16.6T + 79T^{2} \) |
| 83 | \( 1 - 0.327T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.35964226076861006575224970521, −6.85866743971303162731683467861, −5.97959702971433001672782027516, −5.57132214752549197260505751955, −4.88272987558452161347302552977, −4.18652621924215995920433663246, −2.91039198039734957175651547759, −2.17991444279078248192030959561, −1.39771375460542774977146815250, 0,
1.39771375460542774977146815250, 2.17991444279078248192030959561, 2.91039198039734957175651547759, 4.18652621924215995920433663246, 4.88272987558452161347302552977, 5.57132214752549197260505751955, 5.97959702971433001672782027516, 6.85866743971303162731683467861, 7.35964226076861006575224970521
