| L(s) = 1 | + 0.769·2-s + 2.44·3-s − 1.40·4-s − 5-s + 1.88·6-s + 2.92·7-s − 2.62·8-s + 2.98·9-s − 0.769·10-s − 11-s − 3.44·12-s − 6.32·13-s + 2.24·14-s − 2.44·15-s + 0.801·16-s − 1.39·17-s + 2.29·18-s − 5.55·19-s + 1.40·20-s + 7.14·21-s − 0.769·22-s + 9.46·23-s − 6.41·24-s + 25-s − 4.86·26-s − 0.0410·27-s − 4.11·28-s + ⋯ |
| L(s) = 1 | + 0.543·2-s + 1.41·3-s − 0.704·4-s − 0.447·5-s + 0.767·6-s + 1.10·7-s − 0.926·8-s + 0.994·9-s − 0.243·10-s − 0.301·11-s − 0.994·12-s − 1.75·13-s + 0.600·14-s − 0.631·15-s + 0.200·16-s − 0.338·17-s + 0.540·18-s − 1.27·19-s + 0.314·20-s + 1.56·21-s − 0.163·22-s + 1.97·23-s − 1.30·24-s + 0.200·25-s − 0.953·26-s − 0.00789·27-s − 0.778·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 0.769T + 2T^{2} \) |
| 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 5.55T + 19T^{2} \) |
| 23 | \( 1 - 9.46T + 23T^{2} \) |
| 29 | \( 1 + 6.99T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 + 7.40T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 7.08T + 47T^{2} \) |
| 53 | \( 1 + 1.18T + 53T^{2} \) |
| 59 | \( 1 + 4.86T + 59T^{2} \) |
| 61 | \( 1 - 6.73T + 61T^{2} \) |
| 67 | \( 1 + 4.13T + 67T^{2} \) |
| 71 | \( 1 + 0.638T + 71T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 6.34T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.144523864044251007678519083459, −7.52484800161875279167946795366, −6.90408759470793409157834512245, −5.45118010204616269677489884591, −4.85974989933700006999945136485, −4.30137670569247094379546526343, −3.42235422187694266894919888568, −2.65318356784443363642341389904, −1.80929278931363967374907765054, 0,
1.80929278931363967374907765054, 2.65318356784443363642341389904, 3.42235422187694266894919888568, 4.30137670569247094379546526343, 4.85974989933700006999945136485, 5.45118010204616269677489884591, 6.90408759470793409157834512245, 7.52484800161875279167946795366, 8.144523864044251007678519083459
