| L(s) = 1 | − 2.56·2-s + 0.368·3-s + 4.59·4-s − 0.479·5-s − 0.947·6-s + 4.01·7-s − 6.66·8-s − 2.86·9-s + 1.23·10-s + 2.41·11-s + 1.69·12-s + 13-s − 10.3·14-s − 0.176·15-s + 7.93·16-s − 0.131·17-s + 7.35·18-s − 3.90·19-s − 2.20·20-s + 1.48·21-s − 6.19·22-s + 6.17·23-s − 2.45·24-s − 4.77·25-s − 2.56·26-s − 2.16·27-s + 18.4·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.212·3-s + 2.29·4-s − 0.214·5-s − 0.386·6-s + 1.51·7-s − 2.35·8-s − 0.954·9-s + 0.389·10-s + 0.726·11-s + 0.489·12-s + 0.277·13-s − 2.75·14-s − 0.0456·15-s + 1.98·16-s − 0.0319·17-s + 1.73·18-s − 0.896·19-s − 0.492·20-s + 0.323·21-s − 1.32·22-s + 1.28·23-s − 0.502·24-s − 0.954·25-s − 0.503·26-s − 0.416·27-s + 3.48·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 0.368T + 3T^{2} \) |
| 5 | \( 1 + 0.479T + 5T^{2} \) |
| 7 | \( 1 - 4.01T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 17 | \( 1 + 0.131T + 17T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 + 1.74T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 - 5.17T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 - 6.79T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + 7.74T + 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 - 0.0733T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.14T + 71T^{2} \) |
| 73 | \( 1 + 3.87T + 73T^{2} \) |
| 79 | \( 1 + 3.76T + 79T^{2} \) |
| 83 | \( 1 + 6.77T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
| 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
−7.73812715507200947766271927928, −7.19125972256589780919420178585, −6.33779195382897335027103101393, −5.65750392908845407709059491728, −4.69276785875258956318003588683, −3.71052893854548584580322894616, −2.67136339686715135878255525155, −1.86603426688485033134320020221, −1.22542430919594037460712627449, 0,
1.22542430919594037460712627449, 1.86603426688485033134320020221, 2.67136339686715135878255525155, 3.71052893854548584580322894616, 4.69276785875258956318003588683, 5.65750392908845407709059491728, 6.33779195382897335027103101393, 7.19125972256589780919420178585, 7.73812715507200947766271927928
