| L(s) = 1 | − 3-s + 3.56·5-s − 0.955·7-s + 9-s − 5.80·11-s + 2.78·13-s − 3.56·15-s − 4.80·17-s − 2.83·19-s + 0.955·21-s + 23-s + 7.70·25-s − 27-s + 29-s + 1.35·31-s + 5.80·33-s − 3.40·35-s − 2.67·37-s − 2.78·39-s + 10.9·41-s − 5.88·43-s + 3.56·45-s − 0.527·47-s − 6.08·49-s + 4.80·51-s − 7.00·53-s − 20.6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.59·5-s − 0.361·7-s + 0.333·9-s − 1.74·11-s + 0.772·13-s − 0.920·15-s − 1.16·17-s − 0.650·19-s + 0.208·21-s + 0.208·23-s + 1.54·25-s − 0.192·27-s + 0.185·29-s + 0.243·31-s + 1.00·33-s − 0.575·35-s − 0.440·37-s − 0.446·39-s + 1.70·41-s − 0.898·43-s + 0.531·45-s − 0.0768·47-s − 0.869·49-s + 0.672·51-s − 0.962·53-s − 2.78·55-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
| \(L(1)\) |
\(\approx\) |
\(1.711016509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.711016509\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.955T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 13 | \( 1 - 2.78T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 + 2.83T + 19T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.88T + 43T^{2} \) |
| 47 | \( 1 + 0.527T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 - 8.42T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 5.58T + 73T^{2} \) |
| 79 | \( 1 + 9.15T + 79T^{2} \) |
| 83 | \( 1 - 5.98T + 83T^{2} \) |
| 89 | \( 1 - 8.05T + 89T^{2} \) |
| 97 | \( 1 - 8.87T + 97T^{2} \) |
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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
−7.87082621819494647595874525369, −6.82286997447650352540192373691, −6.38107952843305243345687665622, −5.80337665983159578117329687113, −5.14882152955329984324085784073, −4.59673244935518747303715628863, −3.39718045933901772110902647261, −2.41406707019870538441677920794, −1.95767708142481878546483232284, −0.64384141394601771768198564137,
0.64384141394601771768198564137, 1.95767708142481878546483232284, 2.41406707019870538441677920794, 3.39718045933901772110902647261, 4.59673244935518747303715628863, 5.14882152955329984324085784073, 5.80337665983159578117329687113, 6.38107952843305243345687665622, 6.82286997447650352540192373691, 7.87082621819494647595874525369
