L(s) = 1 | − 3-s + 2.82·5-s + 9-s + 2.82·11-s − 2.82·15-s − 2.82·17-s − 4·19-s − 8.48·23-s + 3.00·25-s − 27-s + 2·29-s − 2.82·33-s − 6·37-s − 8.48·41-s − 11.3·43-s + 2.82·45-s − 8·47-s + 2.82·51-s + 6·53-s + 8.00·55-s + 4·57-s − 12·59-s − 5.65·61-s + 5.65·67-s + 8.48·69-s − 2.82·71-s + 5.65·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.26·5-s + 0.333·9-s + 0.852·11-s − 0.730·15-s − 0.685·17-s − 0.917·19-s − 1.76·23-s + 0.600·25-s − 0.192·27-s + 0.371·29-s − 0.492·33-s − 0.986·37-s − 1.32·41-s − 1.72·43-s + 0.421·45-s − 1.16·47-s + 0.396·51-s + 0.824·53-s + 1.07·55-s + 0.529·57-s − 1.56·59-s − 0.724·61-s + 0.691·67-s + 1.02·69-s − 0.335·71-s + 0.662·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.045317270904407179168127782671, −6.81189191556648109007293423016, −6.47078291774047370127331727952, −5.88966250960073648159509475387, −5.05800274523150911462167768773, −4.32074649648640119590403185364, −3.38355135209059472188426201663, −2.04320818966759348032664634680, −1.63211381272935732782874501643, 0,
1.63211381272935732782874501643, 2.04320818966759348032664634680, 3.38355135209059472188426201663, 4.32074649648640119590403185364, 5.05800274523150911462167768773, 5.88966250960073648159509475387, 6.47078291774047370127331727952, 6.81189191556648109007293423016, 8.045317270904407179168127782671