| L(s) = 1 | + 0.696·2-s − 0.956·3-s − 1.51·4-s − 5-s − 0.666·6-s + 7-s − 2.44·8-s − 2.08·9-s − 0.696·10-s − 6.06·11-s + 1.44·12-s − 2.21·13-s + 0.696·14-s + 0.956·15-s + 1.32·16-s + 1.56·17-s − 1.45·18-s + 3.92·19-s + 1.51·20-s − 0.956·21-s − 4.22·22-s + 0.455·23-s + 2.34·24-s + 25-s − 1.54·26-s + 4.86·27-s − 1.51·28-s + ⋯ |
| L(s) = 1 | + 0.492·2-s − 0.552·3-s − 0.757·4-s − 0.447·5-s − 0.272·6-s + 0.377·7-s − 0.865·8-s − 0.695·9-s − 0.220·10-s − 1.82·11-s + 0.418·12-s − 0.615·13-s + 0.186·14-s + 0.246·15-s + 0.330·16-s + 0.378·17-s − 0.342·18-s + 0.899·19-s + 0.338·20-s − 0.208·21-s − 0.901·22-s + 0.0949·23-s + 0.478·24-s + 0.200·25-s − 0.303·26-s + 0.935·27-s − 0.286·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.696T + 2T^{2} \) |
| 3 | \( 1 + 0.956T + 3T^{2} \) |
| 11 | \( 1 + 6.06T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 1.56T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 - 0.455T + 23T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 - 3.89T + 31T^{2} \) |
| 37 | \( 1 - 0.272T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.79T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 + 8.00T + 71T^{2} \) |
| 73 | \( 1 + 3.88T + 73T^{2} \) |
| 79 | \( 1 + 1.54T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−7.63552264652947651467314864278, −6.72857626065704302104394774309, −5.65373735980602257611980264915, −5.41141200245316007484891649537, −4.80148238219291553392997565725, −4.14619955667154931574975251921, −2.92321621245774231600640253162, −2.72341220844648628388079321711, −0.908693685956817356282104171335, 0,
0.908693685956817356282104171335, 2.72341220844648628388079321711, 2.92321621245774231600640253162, 4.14619955667154931574975251921, 4.80148238219291553392997565725, 5.41141200245316007484891649537, 5.65373735980602257611980264915, 6.72857626065704302104394774309, 7.63552264652947651467314864278
