| L(s) = 1 | − 1.15·2-s − 8.67·3-s − 6.67·4-s + 8.17·5-s + 9.98·6-s + 16.8·8-s + 48.2·9-s − 9.41·10-s + 45.1·11-s + 57.9·12-s − 33.5·13-s − 70.9·15-s + 33.9·16-s − 77.4·17-s − 55.5·18-s + 15.9·19-s − 54.5·20-s − 51.9·22-s + 203.·23-s − 146.·24-s − 58.1·25-s + 38.6·26-s − 184.·27-s − 73.8·29-s + 81.6·30-s − 19.9·31-s − 174.·32-s + ⋯ |
| L(s) = 1 | − 0.407·2-s − 1.66·3-s − 0.834·4-s + 0.731·5-s + 0.679·6-s + 0.746·8-s + 1.78·9-s − 0.297·10-s + 1.23·11-s + 1.39·12-s − 0.716·13-s − 1.22·15-s + 0.530·16-s − 1.10·17-s − 0.727·18-s + 0.192·19-s − 0.610·20-s − 0.503·22-s + 1.84·23-s − 1.24·24-s − 0.465·25-s + 0.291·26-s − 1.31·27-s − 0.472·29-s + 0.497·30-s − 0.115·31-s − 0.962·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 1.15T + 8T^{2} \) |
| 3 | \( 1 + 8.67T + 27T^{2} \) |
| 5 | \( 1 - 8.17T + 125T^{2} \) |
| 11 | \( 1 - 45.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 73.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 19.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 135.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 140.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 183.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 26.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 209.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 412.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 706.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 448.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−8.756223858268589116901800829919, −7.33725330653367362973389261207, −6.77258685069482353572744708455, −5.99284092067703310702260968820, −5.10436293249527075892445574833, −4.72350770119987202515473260248, −3.65996667852192402938048020540, −1.86539731033374472338833565823, −0.958342792757913438540803667865, 0,
0.958342792757913438540803667865, 1.86539731033374472338833565823, 3.65996667852192402938048020540, 4.72350770119987202515473260248, 5.10436293249527075892445574833, 5.99284092067703310702260968820, 6.77258685069482353572744708455, 7.33725330653367362973389261207, 8.756223858268589116901800829919
