| L(s) = 1 | + 3-s + 3.77·5-s − 1.75·7-s + 9-s + 2.41·11-s − 1.58·13-s + 3.77·15-s + 0.811·17-s + 3.41·19-s − 1.75·21-s + 6.93·23-s + 9.26·25-s + 27-s − 0.520·29-s − 3.53·31-s + 2.41·33-s − 6.62·35-s − 9.20·37-s − 1.58·39-s − 3.31·41-s + 6.00·43-s + 3.77·45-s + 12.4·47-s − 3.92·49-s + 0.811·51-s + 12.1·53-s + 9.11·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.68·5-s − 0.662·7-s + 0.333·9-s + 0.728·11-s − 0.438·13-s + 0.975·15-s + 0.196·17-s + 0.784·19-s − 0.382·21-s + 1.44·23-s + 1.85·25-s + 0.192·27-s − 0.0966·29-s − 0.635·31-s + 0.420·33-s − 1.11·35-s − 1.51·37-s − 0.253·39-s − 0.517·41-s + 0.914·43-s + 0.563·45-s + 1.81·47-s − 0.560·49-s + 0.113·51-s + 1.66·53-s + 1.22·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.416868696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.416868696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 + 1.75T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 0.811T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 + 0.520T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 + 9.20T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 6.00T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.129T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.920039237099945941505834233887, −7.54711102239799772564292166954, −6.96203252147352321394541286500, −6.26021354633434836695059661854, −5.51915045517375448505896436759, −4.84002785538408413119252205147, −3.60111901124949419337762953508, −2.88987702771623486272092592498, −2.02677107435435936677294233700, −1.11101240598675156660724598223,
1.11101240598675156660724598223, 2.02677107435435936677294233700, 2.88987702771623486272092592498, 3.60111901124949419337762953508, 4.84002785538408413119252205147, 5.51915045517375448505896436759, 6.26021354633434836695059661854, 6.96203252147352321394541286500, 7.54711102239799772564292166954, 8.920039237099945941505834233887
