L(s) = 1 | − 2-s + 0.396·3-s + 4-s − 1.85·5-s − 0.396·6-s + 7-s − 8-s − 2.84·9-s + 1.85·10-s + 4.50·11-s + 0.396·12-s − 5.91·13-s − 14-s − 0.734·15-s + 16-s − 7.27·17-s + 2.84·18-s + 8.32·19-s − 1.85·20-s + 0.396·21-s − 4.50·22-s + 4.63·23-s − 0.396·24-s − 1.56·25-s + 5.91·26-s − 2.31·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.228·3-s + 0.5·4-s − 0.828·5-s − 0.161·6-s + 0.377·7-s − 0.353·8-s − 0.947·9-s + 0.586·10-s + 1.35·11-s + 0.114·12-s − 1.63·13-s − 0.267·14-s − 0.189·15-s + 0.250·16-s − 1.76·17-s + 0.670·18-s + 1.91·19-s − 0.414·20-s + 0.0865·21-s − 0.959·22-s + 0.965·23-s − 0.0809·24-s − 0.313·25-s + 1.15·26-s − 0.445·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9267797108\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9267797108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 0.396T + 3T^{2} \) |
| 5 | \( 1 + 1.85T + 5T^{2} \) |
| 11 | \( 1 - 4.50T + 11T^{2} \) |
| 13 | \( 1 + 5.91T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 - 8.32T + 19T^{2} \) |
| 23 | \( 1 - 4.63T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 + 2.20T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 - 0.513T + 41T^{2} \) |
| 43 | \( 1 + 9.72T + 43T^{2} \) |
| 47 | \( 1 + 2.59T + 47T^{2} \) |
| 53 | \( 1 - 0.359T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 2.79T + 71T^{2} \) |
| 73 | \( 1 + 7.79T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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
−8.119433414867692572956221296028, −7.41829411105630886033293291602, −6.94661532284050644407635818741, −6.16083269884354791852636128820, −5.06375011829048943350504896865, −4.51540398931318039586357122487, −3.43348428913315061118909123357, −2.76806663067023370134253922123, −1.77713145416430200643034026332, −0.54638745010125673762266356771,
0.54638745010125673762266356771, 1.77713145416430200643034026332, 2.76806663067023370134253922123, 3.43348428913315061118909123357, 4.51540398931318039586357122487, 5.06375011829048943350504896865, 6.16083269884354791852636128820, 6.94661532284050644407635818741, 7.41829411105630886033293291602, 8.119433414867692572956221296028