| L(s) = 1 | − 3-s + 4.02·5-s + 4.04·7-s + 9-s − 3.42·11-s + 4.92·13-s − 4.02·15-s + 4.60·17-s + 7.63·19-s − 4.04·21-s + 23-s + 11.1·25-s − 27-s + 29-s − 4.21·31-s + 3.42·33-s + 16.2·35-s + 3.25·37-s − 4.92·39-s − 3.81·41-s − 4.87·43-s + 4.02·45-s + 4.28·47-s + 9.34·49-s − 4.60·51-s + 11.5·53-s − 13.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.79·5-s + 1.52·7-s + 0.333·9-s − 1.03·11-s + 1.36·13-s − 1.03·15-s + 1.11·17-s + 1.75·19-s − 0.882·21-s + 0.208·23-s + 2.23·25-s − 0.192·27-s + 0.185·29-s − 0.757·31-s + 0.596·33-s + 2.74·35-s + 0.534·37-s − 0.787·39-s − 0.595·41-s − 0.744·43-s + 0.599·45-s + 0.624·47-s + 1.33·49-s − 0.645·51-s + 1.59·53-s − 1.85·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.626796886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.626796886\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 4.02T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 4.92T + 13T^{2} \) |
| 17 | \( 1 - 4.60T + 17T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 7.01T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 + 0.476T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−7.69858652208651648528463479661, −7.24833268111680647110724985910, −6.11597673078849600778084527576, −5.63116485508363675261230632942, −5.29480059734058435585044076329, −4.66060105798846080200767251396, −3.40711390130569566113969744040, −2.51483989527770403602132629746, −1.44035737178079189242228739008, −1.21068728938508431452046113396,
1.21068728938508431452046113396, 1.44035737178079189242228739008, 2.51483989527770403602132629746, 3.40711390130569566113969744040, 4.66060105798846080200767251396, 5.29480059734058435585044076329, 5.63116485508363675261230632942, 6.11597673078849600778084527576, 7.24833268111680647110724985910, 7.69858652208651648528463479661
