| L(s) = 1 | − 1.57·2-s + 0.487·4-s + 5.00·7-s + 2.38·8-s + 5.58·11-s + 3.35·13-s − 7.89·14-s − 4.73·16-s + 0.280·17-s − 5.64·19-s − 8.80·22-s + 1.14·23-s − 5.28·26-s + 2.44·28-s − 4.93·29-s + 31-s + 2.70·32-s − 0.442·34-s + 11.1·37-s + 8.90·38-s − 3.66·41-s + 8.88·43-s + 2.72·44-s − 1.81·46-s − 5.65·47-s + 18.0·49-s + 1.63·52-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.243·4-s + 1.89·7-s + 0.843·8-s + 1.68·11-s + 0.930·13-s − 2.11·14-s − 1.18·16-s + 0.0680·17-s − 1.29·19-s − 1.87·22-s + 0.239·23-s − 1.03·26-s + 0.461·28-s − 0.916·29-s + 0.179·31-s + 0.477·32-s − 0.0759·34-s + 1.83·37-s + 1.44·38-s − 0.571·41-s + 1.35·43-s + 0.410·44-s − 0.266·46-s − 0.825·47-s + 2.58·49-s + 0.226·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692085826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692085826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 7 | \( 1 - 5.00T + 7T^{2} \) |
| 11 | \( 1 - 5.58T + 11T^{2} \) |
| 13 | \( 1 - 3.35T + 13T^{2} \) |
| 17 | \( 1 - 0.280T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 23 | \( 1 - 1.14T + 23T^{2} \) |
| 29 | \( 1 + 4.93T + 29T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 3.66T + 41T^{2} \) |
| 43 | \( 1 - 8.88T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4.13T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.50T + 61T^{2} \) |
| 67 | \( 1 + 2.37T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 1.81T + 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 - 9.58T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−8.009811625626859464268987042195, −7.63880657305852718339349321150, −6.67948022342453658729709902629, −6.03842536242983422320270026743, −4.99003600422409612793977661228, −4.28796943840739231297520109788, −3.85476416609053147458694613751, −2.20603771628271852214101239932, −1.51164055926130167531557205690, −0.905858886281876971851797416906,
0.905858886281876971851797416906, 1.51164055926130167531557205690, 2.20603771628271852214101239932, 3.85476416609053147458694613751, 4.28796943840739231297520109788, 4.99003600422409612793977661228, 6.03842536242983422320270026743, 6.67948022342453658729709902629, 7.63880657305852718339349321150, 8.009811625626859464268987042195
