| L(s) = 1 | + 3-s + 2·5-s + 9-s − 4·11-s − 13-s + 2·15-s − 6·17-s + 8·19-s + 8·23-s − 25-s + 27-s − 2·29-s + 8·31-s − 4·33-s + 10·37-s − 39-s + 6·41-s + 4·43-s + 2·45-s − 7·49-s − 6·51-s + 14·53-s − 8·55-s + 8·57-s + 12·59-s + 10·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.516·15-s − 1.45·17-s + 1.83·19-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 + 1.64·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.298·45-s − 49-s − 0.840·51-s + 1.92·53-s − 1.07·55-s + 1.05·57-s + 1.56·59-s + 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.589358492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.589358492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.010117339441672649891089311783, −8.188343340471922629127800088481, −7.40276848314548708606283973789, −6.74853773234864204625876687463, −5.66566431334626487972062313252, −5.09212707298900716958728690515, −4.13812945030733959698956920979, −2.73905592142337669314256041545, −2.49862473157215028411984325777, −1.03001095842298099455230969034,
1.03001095842298099455230969034, 2.49862473157215028411984325777, 2.73905592142337669314256041545, 4.13812945030733959698956920979, 5.09212707298900716958728690515, 5.66566431334626487972062313252, 6.74853773234864204625876687463, 7.40276848314548708606283973789, 8.188343340471922629127800088481, 9.010117339441672649891089311783
