| L(s) = 1 | − 3-s + 3.20·5-s + 9-s + 4.24·11-s + 3.15·13-s − 3.20·15-s + 4.40·17-s − 3.15·19-s + 4.40·23-s + 5.24·25-s − 27-s + 7.20·29-s − 2.04·31-s − 4.24·33-s − 9.65·37-s − 3.15·39-s + 10.4·41-s + 0.750·43-s + 3.20·45-s + 2.40·47-s − 4.40·51-s − 3.29·53-s + 13.6·55-s + 3.15·57-s − 8.24·59-s + 8.09·61-s + 10.0·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.43·5-s + 0.333·9-s + 1.28·11-s + 0.874·13-s − 0.826·15-s + 1.06·17-s − 0.723·19-s + 0.918·23-s + 1.04·25-s − 0.192·27-s + 1.33·29-s − 0.367·31-s − 0.739·33-s − 1.58·37-s − 0.504·39-s + 1.63·41-s + 0.114·43-s + 0.477·45-s + 0.350·47-s − 0.616·51-s − 0.452·53-s + 1.83·55-s + 0.417·57-s − 1.07·59-s + 1.03·61-s + 1.25·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.690981204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.690981204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.20T + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 0.750T + 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 71 | \( 1 - 6.40T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 16.4T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 2.40T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.611565043131948165928423121530, −7.34147004518892955097842480749, −6.64637730683348267657902414434, −6.05820600009165141562909258339, −5.60748839762312218104233348494, −4.70157213118691035747950640750, −3.82214350692409578491386074162, −2.82883155546722697079469801715, −1.64419091416407840581965803806, −1.06390329268472337533951301534,
1.06390329268472337533951301534, 1.64419091416407840581965803806, 2.82883155546722697079469801715, 3.82214350692409578491386074162, 4.70157213118691035747950640750, 5.60748839762312218104233348494, 6.05820600009165141562909258339, 6.64637730683348267657902414434, 7.34147004518892955097842480749, 8.611565043131948165928423121530
