L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4.34·7-s − 8-s + 9-s − 1.07·11-s + 12-s + 2.34·13-s + 4.34·14-s + 16-s − 0.921·17-s − 18-s + 2.34·19-s − 4.34·21-s + 1.07·22-s − 23-s − 24-s − 2.34·26-s + 27-s − 4.34·28-s + 10.4·29-s − 4·31-s − 32-s − 1.07·33-s + 0.921·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.64·7-s − 0.353·8-s + 0.333·9-s − 0.325·11-s + 0.288·12-s + 0.649·13-s + 1.15·14-s + 0.250·16-s − 0.223·17-s − 0.235·18-s + 0.536·19-s − 0.947·21-s + 0.229·22-s − 0.208·23-s − 0.204·24-s − 0.458·26-s + 0.192·27-s − 0.820·28-s + 1.94·29-s − 0.718·31-s − 0.176·32-s − 0.187·33-s + 0.158·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 + 0.921T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 0.156T + 41T^{2} \) |
| 43 | \( 1 + 0.738T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 0.340T + 53T^{2} \) |
| 59 | \( 1 + 8.83T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 2.58T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 4.68T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 + 9.02T + 97T^{2} \) |
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
−8.321508677676329484463199499517, −7.60506886814523222289369793541, −6.68180735587254452461593073657, −6.36346294452675799415739035381, −5.32076779685019756064998667478, −4.08303523085363037730409887917, −3.19016260270778973526500073578, −2.69632030699141062047286179774, −1.36373858304239881915024214646, 0,
1.36373858304239881915024214646, 2.69632030699141062047286179774, 3.19016260270778973526500073578, 4.08303523085363037730409887917, 5.32076779685019756064998667478, 6.36346294452675799415739035381, 6.68180735587254452461593073657, 7.60506886814523222289369793541, 8.321508677676329484463199499517