| L(s) = 1 | + 2.76·2-s − 3.22·3-s + 5.65·4-s − 5-s − 8.93·6-s − 2.80·7-s + 10.1·8-s + 7.42·9-s − 2.76·10-s − 11-s − 18.2·12-s + 0.723·13-s − 7.75·14-s + 3.22·15-s + 16.6·16-s − 6.98·17-s + 20.5·18-s − 1.18·19-s − 5.65·20-s + 9.04·21-s − 2.76·22-s + 3.84·23-s − 32.6·24-s + 25-s + 2.00·26-s − 14.2·27-s − 15.8·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 1.86·3-s + 2.82·4-s − 0.447·5-s − 3.64·6-s − 1.05·7-s + 3.57·8-s + 2.47·9-s − 0.875·10-s − 0.301·11-s − 5.27·12-s + 0.200·13-s − 2.07·14-s + 0.833·15-s + 4.17·16-s − 1.69·17-s + 4.84·18-s − 0.271·19-s − 1.26·20-s + 1.97·21-s − 0.589·22-s + 0.801·23-s − 6.67·24-s + 0.200·25-s + 0.392·26-s − 2.74·27-s − 2.99·28-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 | \( 1 - 2.76T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 13 | \( 1 - 0.723T + 13T^{2} \) |
| 17 | \( 1 + 6.98T + 17T^{2} \) |
| 19 | \( 1 + 1.18T + 19T^{2} \) |
| 23 | \( 1 - 3.84T + 23T^{2} \) |
| 29 | \( 1 - 0.803T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 0.991T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 3.30T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 7.63T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 79 | \( 1 + 8.54T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 1.57T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−7.30322077144275720769091394266, −6.82873361547339823166062045897, −6.45385887345253596146170036559, −5.76019001660946841755452468609, −5.02312591011139581012702107357, −4.50695404466774316381404503966, −3.77870514910261378878790077562, −2.84979007397354309929980484147, −1.59077696740748948376915317944, 0,
1.59077696740748948376915317944, 2.84979007397354309929980484147, 3.77870514910261378878790077562, 4.50695404466774316381404503966, 5.02312591011139581012702107357, 5.76019001660946841755452468609, 6.45385887345253596146170036559, 6.82873361547339823166062045897, 7.30322077144275720769091394266
