| L(s) = 1 | + 1.35·2-s − 3.16·3-s − 0.173·4-s + 0.946·5-s − 4.28·6-s − 1.01·7-s − 2.93·8-s + 7.03·9-s + 1.27·10-s + 11-s + 0.549·12-s − 0.670·13-s − 1.37·14-s − 2.99·15-s − 3.62·16-s − 17-s + 9.51·18-s + 7.01·19-s − 0.164·20-s + 3.21·21-s + 1.35·22-s + 5.30·23-s + 9.30·24-s − 4.10·25-s − 0.905·26-s − 12.7·27-s + 0.176·28-s + ⋯ |
| L(s) = 1 | + 0.955·2-s − 1.82·3-s − 0.0867·4-s + 0.423·5-s − 1.74·6-s − 0.384·7-s − 1.03·8-s + 2.34·9-s + 0.404·10-s + 0.301·11-s + 0.158·12-s − 0.185·13-s − 0.367·14-s − 0.774·15-s − 0.905·16-s − 0.242·17-s + 2.24·18-s + 1.60·19-s − 0.0367·20-s + 0.702·21-s + 0.288·22-s + 1.10·23-s + 1.89·24-s − 0.820·25-s − 0.177·26-s − 2.46·27-s + 0.0333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.246202787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246202787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 0.946T + 5T^{2} \) |
| 7 | \( 1 + 1.01T + 7T^{2} \) |
| 13 | \( 1 + 0.670T + 13T^{2} \) |
| 19 | \( 1 - 7.01T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 1.08T + 31T^{2} \) |
| 37 | \( 1 + 8.33T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 47 | \( 1 + 8.42T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 - 4.60T + 59T^{2} \) |
| 61 | \( 1 - 4.42T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 - 0.284T + 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 + 0.746T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 - 6.02T + 97T^{2} \) |
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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
−7.37090373260867300591719015273, −6.76118495246408396176325776552, −6.27006788376403106320396790921, −5.57492728088834333675102199622, −5.10544781104407484293761641553, −4.65739909945678066285725710168, −3.72294482306746604645048864477, −2.97534681457485368688454010257, −1.55948125601745353069601031689, −0.54821621256652192682241914285,
0.54821621256652192682241914285, 1.55948125601745353069601031689, 2.97534681457485368688454010257, 3.72294482306746604645048864477, 4.65739909945678066285725710168, 5.10544781104407484293761641553, 5.57492728088834333675102199622, 6.27006788376403106320396790921, 6.76118495246408396176325776552, 7.37090373260867300591719015273
