| L(s) = 1 | − 2-s − 4-s + 5-s − 7-s + 3·8-s − 10-s − 6·13-s + 14-s − 16-s − 8·19-s − 20-s + 8·23-s + 25-s + 6·26-s + 28-s − 2·29-s − 4·31-s − 5·32-s − 35-s + 2·37-s + 8·38-s + 3·40-s − 6·41-s + 4·43-s − 8·46-s − 8·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 1.83·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.169·35-s + 0.328·37-s + 1.29·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.22858927482673, −13.45479199108873, −13.01429506164146, −12.76381684087660, −12.34426098021916, −11.46084060496254, −11.02176476619624, −10.42776152391942, −10.09769366826244, −9.577789237620261, −9.100257599021185, −8.796321473426983, −8.168771027468686, −7.514459232833492, −7.161535862642481, −6.563431571603663, −5.989426154280033, −5.200354730721928, −4.758669002335155, −4.425866140315044, −3.510647918394131, −2.916193653727692, −2.105875044818925, −1.690545022315217, −0.6387756184465831, 0,
0.6387756184465831, 1.690545022315217, 2.105875044818925, 2.916193653727692, 3.510647918394131, 4.425866140315044, 4.758669002335155, 5.200354730721928, 5.989426154280033, 6.563431571603663, 7.161535862642481, 7.514459232833492, 8.168771027468686, 8.796321473426983, 9.100257599021185, 9.577789237620261, 10.09769366826244, 10.42776152391942, 11.02176476619624, 11.46084060496254, 12.34426098021916, 12.76381684087660, 13.01429506164146, 13.45479199108873, 14.22858927482673
