| L(s) = 1 | + 2.53·3-s − 0.138·5-s − 7-s + 3.44·9-s − 0.938·11-s − 3.02·13-s − 0.352·15-s + 17-s − 2.38·19-s − 2.53·21-s − 5.51·23-s − 4.98·25-s + 1.11·27-s − 4.99·29-s − 9.85·31-s − 2.38·33-s + 0.138·35-s − 6.85·37-s − 7.68·39-s − 4.61·41-s + 1.86·43-s − 0.477·45-s + 0.273·47-s + 49-s + 2.53·51-s + 6.18·53-s + 0.130·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s − 0.0621·5-s − 0.377·7-s + 1.14·9-s − 0.282·11-s − 0.840·13-s − 0.0910·15-s + 0.242·17-s − 0.546·19-s − 0.553·21-s − 1.15·23-s − 0.996·25-s + 0.215·27-s − 0.927·29-s − 1.76·31-s − 0.414·33-s + 0.0234·35-s − 1.12·37-s − 1.23·39-s − 0.720·41-s + 0.283·43-s − 0.0712·45-s + 0.0398·47-s + 0.142·49-s + 0.355·51-s + 0.849·53-s + 0.0175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 + 0.138T + 5T^{2} \) |
| 11 | \( 1 + 0.938T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 + 4.99T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 4.61T + 41T^{2} \) |
| 43 | \( 1 - 1.86T + 43T^{2} \) |
| 47 | \( 1 - 0.273T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 - 2.88T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 5.96T + 79T^{2} \) |
| 83 | \( 1 + 8.92T + 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.156140563176243480704110830028, −7.53020522000278276109599078214, −6.96624629547728354464213237676, −5.86454094895825218727060994839, −5.08464846160462524505193536880, −3.78388545626032308706676310635, −3.63533281951626489136344682016, −2.35202316011113328905799288168, −1.95396193169749418340536233522, 0,
1.95396193169749418340536233522, 2.35202316011113328905799288168, 3.63533281951626489136344682016, 3.78388545626032308706676310635, 5.08464846160462524505193536880, 5.86454094895825218727060994839, 6.96624629547728354464213237676, 7.53020522000278276109599078214, 8.156140563176243480704110830028
