| L(s) = 1 | − 2.74·2-s − 3-s + 5.53·4-s + 0.596·5-s + 2.74·6-s − 1.94·7-s − 9.71·8-s + 9-s − 1.63·10-s + 0.999·11-s − 5.53·12-s − 13-s + 5.34·14-s − 0.596·15-s + 15.5·16-s − 0.945·17-s − 2.74·18-s + 3.15·19-s + 3.30·20-s + 1.94·21-s − 2.74·22-s − 3.10·23-s + 9.71·24-s − 4.64·25-s + 2.74·26-s − 27-s − 10.7·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.76·4-s + 0.266·5-s + 1.12·6-s − 0.735·7-s − 3.43·8-s + 0.333·9-s − 0.517·10-s + 0.301·11-s − 1.59·12-s − 0.277·13-s + 1.42·14-s − 0.153·15-s + 3.89·16-s − 0.229·17-s − 0.647·18-s + 0.724·19-s + 0.738·20-s + 0.424·21-s − 0.584·22-s − 0.647·23-s + 1.98·24-s − 0.928·25-s + 0.538·26-s − 0.192·27-s − 2.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 5 | \( 1 - 0.596T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 - 0.999T + 11T^{2} \) |
| 17 | \( 1 + 0.945T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 + 7.44T + 29T^{2} \) |
| 31 | \( 1 - 9.61T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 1.29T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 0.571T + 61T^{2} \) |
| 67 | \( 1 + 5.30T + 67T^{2} \) |
| 71 | \( 1 + 3.37T + 71T^{2} \) |
| 73 | \( 1 - 6.15T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 0.0263T + 89T^{2} \) |
| 97 | \( 1 - 2.13T + 97T^{2} \) |
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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
−8.043352690508392834517627576333, −7.50188254143404944175941279316, −6.81736370155536493892593340477, −6.07823782770669005060466005494, −5.65924180971016690094950205655, −4.12143226224310736642937197996, −2.96617790705191884641866225823, −2.10292100523978529663966891276, −1.05945265898304083312506755436, 0,
1.05945265898304083312506755436, 2.10292100523978529663966891276, 2.96617790705191884641866225823, 4.12143226224310736642937197996, 5.65924180971016690094950205655, 6.07823782770669005060466005494, 6.81736370155536493892593340477, 7.50188254143404944175941279316, 8.043352690508392834517627576333
