| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 4·11-s − 15-s + 3·17-s + 4·19-s − 3·21-s − 23-s + 25-s − 27-s + 29-s − 31-s − 4·33-s + 3·35-s + 37-s + 3·41-s + 12·43-s + 45-s + 10·47-s + 2·49-s − 3·51-s − 9·53-s + 4·55-s − 4·57-s + 9·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s − 0.258·15-s + 0.727·17-s + 0.917·19-s − 0.654·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 0.179·31-s − 0.696·33-s + 0.507·35-s + 0.164·37-s + 0.468·41-s + 1.82·43-s + 0.149·45-s + 1.45·47-s + 2/7·49-s − 0.420·51-s − 1.23·53-s + 0.539·55-s − 0.529·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.542183622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.542183622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−8.004589288267435145820470720141, −7.47126786809375702134726962515, −6.69000064429091597350822477494, −5.84937356613425091986945301104, −5.41113840750206704613699327543, −4.51555652660007959623588128865, −3.90174589367570301148880809288, −2.73926659342582591393750611351, −1.59025694250493072573712589851, −0.999253476935963342398791421140,
0.999253476935963342398791421140, 1.59025694250493072573712589851, 2.73926659342582591393750611351, 3.90174589367570301148880809288, 4.51555652660007959623588128865, 5.41113840750206704613699327543, 5.84937356613425091986945301104, 6.69000064429091597350822477494, 7.47126786809375702134726962515, 8.004589288267435145820470720141
