| L(s) = 1 | + 2.58·2-s + 4.65·4-s − 0.947·7-s + 6.86·8-s − 4.90·11-s − 0.508·13-s − 2.44·14-s + 8.39·16-s − 4.98·17-s + 0.580·19-s − 12.6·22-s + 1.54·23-s − 1.31·26-s − 4.41·28-s − 9.55·29-s − 9.75·31-s + 7.92·32-s − 12.8·34-s − 37-s + 1.49·38-s + 0.960·41-s − 0.875·43-s − 22.8·44-s + 3.98·46-s + 8.16·47-s − 6.10·49-s − 2.36·52-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.32·4-s − 0.358·7-s + 2.42·8-s − 1.47·11-s − 0.140·13-s − 0.653·14-s + 2.09·16-s − 1.20·17-s + 0.133·19-s − 2.69·22-s + 0.321·23-s − 0.257·26-s − 0.834·28-s − 1.77·29-s − 1.75·31-s + 1.40·32-s − 2.20·34-s − 0.164·37-s + 0.243·38-s + 0.150·41-s − 0.133·43-s − 3.44·44-s + 0.587·46-s + 1.19·47-s − 0.871·49-s − 0.328·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 7 | \( 1 + 0.947T + 7T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 0.508T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 - 0.580T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 9.55T + 29T^{2} \) |
| 31 | \( 1 + 9.75T + 31T^{2} \) |
| 41 | \( 1 - 0.960T + 41T^{2} \) |
| 43 | \( 1 + 0.875T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 59 | \( 1 - 6.36T + 59T^{2} \) |
| 61 | \( 1 + 0.166T + 61T^{2} \) |
| 67 | \( 1 + 5.85T + 67T^{2} \) |
| 71 | \( 1 - 5.41T + 71T^{2} \) |
| 73 | \( 1 + 3.50T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 6.21T + 97T^{2} \) |
| 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
−7.26657113470292593392223546398, −6.62869996311056989380659379008, −5.69499044664078178064863501644, −5.43566390147651387178069908543, −4.67151118083174044424440562990, −3.93248938191933883368365438910, −3.25789916019970985149241668522, −2.46577579663443202006489286001, −1.88172479868451566796040712222, 0,
1.88172479868451566796040712222, 2.46577579663443202006489286001, 3.25789916019970985149241668522, 3.93248938191933883368365438910, 4.67151118083174044424440562990, 5.43566390147651387178069908543, 5.69499044664078178064863501644, 6.62869996311056989380659379008, 7.26657113470292593392223546398
