| 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
| \(L(1)\) |
\(\approx\) |
\(2.001049487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001049487\) |
| \(L(\frac{3}{2})\) |
|
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
−8.437871548224310202742352185353, −7.949152959733991878916874270549, −7.31979595951381878058034312516, −6.55341822008387752471814319456, −5.36060461079758519453871924476, −4.81491928124302049005015756014, −3.92742709962574332759519972128, −2.88633232471614386188270893470, −2.28839099576648481139527420277, −0.813170101248483559516930675379,
0.813170101248483559516930675379, 2.28839099576648481139527420277, 2.88633232471614386188270893470, 3.92742709962574332759519972128, 4.81491928124302049005015756014, 5.36060461079758519453871924476, 6.55341822008387752471814319456, 7.31979595951381878058034312516, 7.949152959733991878916874270549, 8.437871548224310202742352185353
