| L(s) = 1 | − 0.395·2-s + 2.73·3-s − 1.84·4-s − 1.08·6-s − 0.629·7-s + 1.51·8-s + 4.48·9-s + 5.49·11-s − 5.04·12-s + 0.248·14-s + 3.08·16-s − 4.25·17-s − 1.77·18-s + 3.61·19-s − 1.72·21-s − 2.17·22-s + 7.05·23-s + 4.15·24-s + 4.07·27-s + 1.16·28-s + 2.09·29-s + 2.97·31-s − 4.25·32-s + 15.0·33-s + 1.68·34-s − 8.27·36-s − 0.817·37-s + ⋯ |
| L(s) = 1 | − 0.279·2-s + 1.57·3-s − 0.921·4-s − 0.441·6-s − 0.237·7-s + 0.536·8-s + 1.49·9-s + 1.65·11-s − 1.45·12-s + 0.0664·14-s + 0.771·16-s − 1.03·17-s − 0.417·18-s + 0.829·19-s − 0.375·21-s − 0.463·22-s + 1.47·23-s + 0.848·24-s + 0.783·27-s + 0.219·28-s + 0.389·29-s + 0.534·31-s − 0.752·32-s + 2.61·33-s + 0.288·34-s − 1.37·36-s − 0.134·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.725747521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.725747521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.395T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 0.629T + 7T^{2} \) |
| 11 | \( 1 - 5.49T + 11T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 - 2.09T + 29T^{2} \) |
| 31 | \( 1 - 2.97T + 31T^{2} \) |
| 37 | \( 1 + 0.817T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 - 2.29T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 - 5.92T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 0.210T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 7.19T + 79T^{2} \) |
| 83 | \( 1 - 6.61T + 83T^{2} \) |
| 89 | \( 1 - 3.66T + 89T^{2} \) |
| 97 | \( 1 - 4.54T + 97T^{2} \) |
| 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
−8.550108966613235487100312789172, −8.017804055524007130678260149016, −6.98525730583231855762966062497, −6.58794062497786606101596706015, −5.16784168235767561807234058278, −4.44201657201101523856589188103, −3.63069314103470725965438589703, −3.15671124294263390648149857090, −1.92266447179534043179260802676, −0.975028515360273386705947820931,
0.975028515360273386705947820931, 1.92266447179534043179260802676, 3.15671124294263390648149857090, 3.63069314103470725965438589703, 4.44201657201101523856589188103, 5.16784168235767561807234058278, 6.58794062497786606101596706015, 6.98525730583231855762966062497, 8.017804055524007130678260149016, 8.550108966613235487100312789172
