L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 13-s − 15-s + 2·17-s + 4·19-s − 8·23-s + 25-s − 27-s − 2·29-s + 8·31-s + 4·33-s + 6·37-s − 39-s − 6·41-s + 4·43-s + 45-s + 8·47-s − 7·49-s − 2·51-s + 6·53-s − 4·55-s − 4·57-s + 12·59-s − 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.474116317\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.474116317\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.589432308568120532399394263407, −7.88541805670611931278793272721, −7.25526504565177991185560232138, −6.20235989153436615567604779531, −5.70617428080584220916058512582, −5.00649007098692076039682952910, −4.08370634935231621549895187340, −2.99305738219524609691519223861, −2.03833641567320312947705107247, −0.75387101601756041955604763488,
0.75387101601756041955604763488, 2.03833641567320312947705107247, 2.99305738219524609691519223861, 4.08370634935231621549895187340, 5.00649007098692076039682952910, 5.70617428080584220916058512582, 6.20235989153436615567604779531, 7.25526504565177991185560232138, 7.88541805670611931278793272721, 8.589432308568120532399394263407