| L(s) = 1 | + 0.983·2-s + 3-s − 1.03·4-s + 1.93·5-s + 0.983·6-s + 1.20·7-s − 2.98·8-s + 9-s + 1.90·10-s + 3.67·11-s − 1.03·12-s − 6.12·13-s + 1.18·14-s + 1.93·15-s − 0.865·16-s − 17-s + 0.983·18-s − 2.81·19-s − 1.99·20-s + 1.20·21-s + 3.61·22-s − 3.42·23-s − 2.98·24-s − 1.25·25-s − 6.02·26-s + 27-s − 1.24·28-s + ⋯ |
| L(s) = 1 | + 0.695·2-s + 0.577·3-s − 0.516·4-s + 0.865·5-s + 0.401·6-s + 0.454·7-s − 1.05·8-s + 0.333·9-s + 0.601·10-s + 1.10·11-s − 0.298·12-s − 1.70·13-s + 0.315·14-s + 0.499·15-s − 0.216·16-s − 0.242·17-s + 0.231·18-s − 0.645·19-s − 0.447·20-s + 0.262·21-s + 0.770·22-s − 0.714·23-s − 0.608·24-s − 0.251·25-s − 1.18·26-s + 0.192·27-s − 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.983T + 2T^{2} \) |
| 5 | \( 1 - 1.93T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 19 | \( 1 + 2.81T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 0.456T + 31T^{2} \) |
| 37 | \( 1 + 0.129T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 - 2.29T + 43T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 53 | \( 1 + 7.68T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 0.0902T + 79T^{2} \) |
| 83 | \( 1 + 9.40T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.46715163184530083064279290446, −6.69485480826510565971006390314, −5.95210111274837331343418988981, −5.33961055918770452456920648827, −4.51452378645876974950876043867, −4.11187929735052618584622047534, −3.16370540206860146908485965207, −2.26968935123885978635825928333, −1.62328395581137710762333754652, 0,
1.62328395581137710762333754652, 2.26968935123885978635825928333, 3.16370540206860146908485965207, 4.11187929735052618584622047534, 4.51452378645876974950876043867, 5.33961055918770452456920648827, 5.95210111274837331343418988981, 6.69485480826510565971006390314, 7.46715163184530083064279290446
