| L(s) = 1 | − 2.53·2-s − 1.59·4-s + 27.8·7-s + 24.2·8-s + 35.8·11-s − 27.3·13-s − 70.5·14-s − 48.6·16-s + 93.6·17-s + 135.·19-s − 90.6·22-s − 0.407·23-s + 69.2·26-s − 44.5·28-s + 194.·29-s − 96.7·31-s − 71.1·32-s − 236.·34-s − 186.·37-s − 343.·38-s − 53.6·41-s − 519.·43-s − 57.2·44-s + 1.03·46-s + 190.·47-s + 434.·49-s + 43.7·52-s + ⋯ |
| L(s) = 1 | − 0.894·2-s − 0.199·4-s + 1.50·7-s + 1.07·8-s + 0.981·11-s − 0.583·13-s − 1.34·14-s − 0.760·16-s + 1.33·17-s + 1.63·19-s − 0.878·22-s − 0.00369·23-s + 0.522·26-s − 0.300·28-s + 1.24·29-s − 0.560·31-s − 0.393·32-s − 1.19·34-s − 0.830·37-s − 1.46·38-s − 0.204·41-s − 1.84·43-s − 0.196·44-s + 0.00330·46-s + 0.591·47-s + 1.26·49-s + 0.116·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.564055272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564055272\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.53T + 8T^{2} \) |
| 7 | \( 1 - 27.8T + 343T^{2} \) |
| 11 | \( 1 - 35.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 0.407T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 96.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 186.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 53.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 519.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 533.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 472.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 327.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 78.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 344.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 98.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 448.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
| 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
−9.920279182618299182715821057595, −9.280828050472990049248076615079, −8.297117857678951953725167394289, −7.78598324336757235089341246393, −6.92551060141358386486629757652, −5.33067488385657575166064099733, −4.76502419260906597287252224153, −3.48704778763113943636216315049, −1.69904924127376626688144351769, −0.936627922637311638810939463396,
0.936627922637311638810939463396, 1.69904924127376626688144351769, 3.48704778763113943636216315049, 4.76502419260906597287252224153, 5.33067488385657575166064099733, 6.92551060141358386486629757652, 7.78598324336757235089341246393, 8.297117857678951953725167394289, 9.280828050472990049248076615079, 9.920279182618299182715821057595
