| L(s) = 1 | − 2.49·3-s − 3.92·5-s + 2.18·7-s + 3.21·9-s + 3.47·11-s + 3.85·13-s + 9.78·15-s − 2.14·17-s + 7.07·19-s − 5.44·21-s − 0.572·23-s + 10.4·25-s − 0.530·27-s + 5.29·29-s + 8.09·31-s − 8.66·33-s − 8.58·35-s + 4.15·37-s − 9.61·39-s + 0.245·41-s + 5.43·43-s − 12.6·45-s + 1.50·47-s − 2.22·49-s + 5.35·51-s − 7.38·53-s − 13.6·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s − 1.75·5-s + 0.826·7-s + 1.07·9-s + 1.04·11-s + 1.06·13-s + 2.52·15-s − 0.520·17-s + 1.62·19-s − 1.18·21-s − 0.119·23-s + 2.08·25-s − 0.102·27-s + 0.982·29-s + 1.45·31-s − 1.50·33-s − 1.45·35-s + 0.683·37-s − 1.53·39-s + 0.0382·41-s + 0.828·43-s − 1.88·45-s + 0.218·47-s − 0.317·49-s + 0.749·51-s − 1.01·53-s − 1.84·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124318679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124318679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 3.92T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 0.572T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 0.245T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 7.38T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 - 5.93T + 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 - 0.909T + 73T^{2} \) |
| 79 | \( 1 + 9.07T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.29T + 97T^{2} \) |
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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
−7.996645708832103019326712625195, −7.35771902626778793173709519537, −6.56007737673676964657952142090, −6.08306511768089508634513864709, −5.01132477067290910787592032199, −4.54388180519976692622655625129, −3.91302622248643017202298359618, −3.03876658572444289471213391055, −1.25203912436925629832124333926, −0.73225474071909929962841319570,
0.73225474071909929962841319570, 1.25203912436925629832124333926, 3.03876658572444289471213391055, 3.91302622248643017202298359618, 4.54388180519976692622655625129, 5.01132477067290910787592032199, 6.08306511768089508634513864709, 6.56007737673676964657952142090, 7.35771902626778793173709519537, 7.996645708832103019326712625195
