| L(s) = 1 | − 1.93·2-s + 1.73·4-s − 5-s + 0.517·8-s + 1.93·10-s − 1.03·11-s + 3.86·13-s − 4.46·16-s − 5.46·17-s + 2.44·19-s − 1.73·20-s + 1.99·22-s − 1.41·23-s + 25-s − 7.46·26-s + 2.07·29-s − 2.44·31-s + 7.58·32-s + 10.5·34-s − 8.92·37-s − 4.73·38-s − 0.517·40-s − 7.46·41-s + 12.3·43-s − 1.79·44-s + 2.73·46-s + 0.928·47-s + ⋯ |
| L(s) = 1 | − 1.36·2-s + 0.866·4-s − 0.447·5-s + 0.183·8-s + 0.610·10-s − 0.312·11-s + 1.07·13-s − 1.11·16-s − 1.32·17-s + 0.561·19-s − 0.387·20-s + 0.426·22-s − 0.294·23-s + 0.200·25-s − 1.46·26-s + 0.384·29-s − 0.439·31-s + 1.34·32-s + 1.81·34-s − 1.46·37-s − 0.767·38-s − 0.0818·40-s − 1.16·41-s + 1.88·43-s − 0.270·44-s + 0.402·46-s + 0.135·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 + 5.46T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 2.07T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 0.928T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 9.52T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 0.928T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
| 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
−8.680308992761927953725807510872, −8.160213829625065689857373166060, −7.26166803825357449909146943026, −6.72015404601520075783537090433, −5.63213754738436009364355405987, −4.54832414471839605404309164238, −3.67287801154017199583315878613, −2.38985477759701404734562896432, −1.27288953058477649763091164483, 0,
1.27288953058477649763091164483, 2.38985477759701404734562896432, 3.67287801154017199583315878613, 4.54832414471839605404309164238, 5.63213754738436009364355405987, 6.72015404601520075783537090433, 7.26166803825357449909146943026, 8.160213829625065689857373166060, 8.680308992761927953725807510872
