L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 5·11-s + 4·13-s + 15-s + 2·17-s + 5·19-s − 21-s − 23-s + 25-s + 27-s − 2·29-s + 8·31-s − 5·33-s − 35-s − 6·37-s + 4·39-s − 41-s + 10·43-s + 45-s − 7·47-s + 49-s + 2·51-s + 53-s − 5·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s + 0.258·15-s + 0.485·17-s + 1.14·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.870·33-s − 0.169·35-s − 0.986·37-s + 0.640·39-s − 0.156·41-s + 1.52·43-s + 0.149·45-s − 1.02·47-s + 1/7·49-s + 0.280·51-s + 0.137·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.765645335\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.765645335\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.83869522695765188678932340158, −7.07469723432016191308971408461, −6.29807064253494856277967076127, −5.58860205323663486190317502030, −5.06996226182425459608785193319, −4.07534083867003697812534556946, −3.22106526273465496165860104783, −2.77087814328524870218910301613, −1.79850208517173616283660302640, −0.77774965185840225548707362575,
0.77774965185840225548707362575, 1.79850208517173616283660302640, 2.77087814328524870218910301613, 3.22106526273465496165860104783, 4.07534083867003697812534556946, 5.06996226182425459608785193319, 5.58860205323663486190317502030, 6.29807064253494856277967076127, 7.07469723432016191308971408461, 7.83869522695765188678932340158