| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s − 2·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 25-s − 27-s + 10·29-s − 4·33-s + 35-s + 6·37-s + 2·39-s − 10·41-s − 8·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 6·53-s − 4·55-s + 4·57-s + 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.320·39-s − 1.56·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.398488442216376060800585323034, −7.32412694070645715715841837301, −6.58231640570943695170243461055, −6.29210754378711639558402142799, −5.06313695660856529095288192450, −4.41505341237461983085321975339, −3.65314900404320595304089820888, −2.55858560243792014719014603355, −1.30012330121326572487096070830, 0,
1.30012330121326572487096070830, 2.55858560243792014719014603355, 3.65314900404320595304089820888, 4.41505341237461983085321975339, 5.06313695660856529095288192450, 6.29210754378711639558402142799, 6.58231640570943695170243461055, 7.32412694070645715715841837301, 8.398488442216376060800585323034
