L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 5·11-s + 13-s + 15-s + 3·17-s − 6·19-s + 21-s − 6·23-s + 25-s + 5·27-s − 9·29-s + 5·33-s + 35-s + 6·37-s − 39-s + 8·41-s + 6·43-s + 2·45-s + 3·47-s + 49-s − 3·51-s − 12·53-s + 5·55-s + 6·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 1.67·29-s + 0.870·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s + 1.24·41-s + 0.914·43-s + 0.298·45-s + 0.437·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s + 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20832939807863271504904378806, −10.71481393224656336284578498652, −9.607085486774063138914067404916, −8.321037332106296729315848288758, −7.61068895852562797070369378134, −6.17106315121314565062659046983, −5.45734437575009036836059962399, −4.06182673812785905054132006247, −2.60467600555756229149551481858, 0,
2.60467600555756229149551481858, 4.06182673812785905054132006247, 5.45734437575009036836059962399, 6.17106315121314565062659046983, 7.61068895852562797070369378134, 8.321037332106296729315848288758, 9.607085486774063138914067404916, 10.71481393224656336284578498652, 11.20832939807863271504904378806