| L(s) = 1 | + 2.80·2-s − 3-s + 5.84·4-s + 2.43·5-s − 2.80·6-s + 7-s + 10.7·8-s + 9-s + 6.81·10-s − 1.69·11-s − 5.84·12-s + 1.08·13-s + 2.80·14-s − 2.43·15-s + 18.4·16-s + 1.83·17-s + 2.80·18-s + 2.78·19-s + 14.2·20-s − 21-s − 4.75·22-s − 8.57·23-s − 10.7·24-s + 0.925·25-s + 3.02·26-s − 27-s + 5.84·28-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 0.577·3-s + 2.92·4-s + 1.08·5-s − 1.14·6-s + 0.377·7-s + 3.80·8-s + 0.333·9-s + 2.15·10-s − 0.512·11-s − 1.68·12-s + 0.299·13-s + 0.748·14-s − 0.628·15-s + 4.61·16-s + 0.443·17-s + 0.660·18-s + 0.638·19-s + 3.18·20-s − 0.218·21-s − 1.01·22-s − 1.78·23-s − 2.19·24-s + 0.185·25-s + 0.593·26-s − 0.192·27-s + 1.10·28-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
| \(L(1)\) |
\(\approx\) |
\(7.993303442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.993303442\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.80T + 2T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 1.08T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 + 8.57T + 23T^{2} \) |
| 29 | \( 1 + 7.87T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 - 2.79T + 41T^{2} \) |
| 43 | \( 1 - 6.69T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 + 8.57T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 + 3.95T + 79T^{2} \) |
| 83 | \( 1 + 7.47T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.86014995832004611379190532147, −7.58245455040963485366527641262, −6.37011529642762996700303086515, −6.03974488680200073493353982726, −5.47635058228508086651194476053, −4.83003039654200542357923717299, −4.03975200751144489236357293189, −3.13328516992549653119516887040, −2.14420427455433082153550856086, −1.49434823894810995541398945881,
1.49434823894810995541398945881, 2.14420427455433082153550856086, 3.13328516992549653119516887040, 4.03975200751144489236357293189, 4.83003039654200542357923717299, 5.47635058228508086651194476053, 6.03974488680200073493353982726, 6.37011529642762996700303086515, 7.58245455040963485366527641262, 7.86014995832004611379190532147
