| L(s) = 1 | + 2.73·2-s − 3-s + 5.46·4-s − 3.27·5-s − 2.73·6-s − 7-s + 9.45·8-s + 9-s − 8.93·10-s − 0.282·11-s − 5.46·12-s + 3.54·13-s − 2.73·14-s + 3.27·15-s + 14.9·16-s + 2.73·17-s + 2.73·18-s − 2.06·19-s − 17.8·20-s + 21-s − 0.771·22-s − 0.194·23-s − 9.45·24-s + 5.69·25-s + 9.67·26-s − 27-s − 5.46·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.577·3-s + 2.73·4-s − 1.46·5-s − 1.11·6-s − 0.377·7-s + 3.34·8-s + 0.333·9-s − 2.82·10-s − 0.0851·11-s − 1.57·12-s + 0.982·13-s − 0.730·14-s + 0.844·15-s + 3.72·16-s + 0.662·17-s + 0.643·18-s − 0.472·19-s − 3.99·20-s + 0.218·21-s − 0.164·22-s − 0.0404·23-s − 1.93·24-s + 1.13·25-s + 1.89·26-s − 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.605693748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.605693748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + 0.282T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 23 | \( 1 + 0.194T + 23T^{2} \) |
| 29 | \( 1 + 0.879T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 6.56T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 - 9.85T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 7.15T + 83T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.908909567556873022193621773744, −7.50320166727095660219745498373, −6.75177396414332554903808986264, −5.95411453308027001446189129142, −5.52053145530676108103175141254, −4.38960927353426257227274799582, −4.04403646958356023345789507317, −3.40242669105301495835238427125, −2.44627812507464745451162414134, −0.973491608643606770653151852725,
0.973491608643606770653151852725, 2.44627812507464745451162414134, 3.40242669105301495835238427125, 4.04403646958356023345789507317, 4.38960927353426257227274799582, 5.52053145530676108103175141254, 5.95411453308027001446189129142, 6.75177396414332554903808986264, 7.50320166727095660219745498373, 7.908909567556873022193621773744
