| L(s) = 1 | − 14·5-s + 170·7-s − 243·9-s + 250·11-s − 169·13-s + 1.06e3·17-s + 78·19-s − 1.57e3·23-s − 2.92e3·25-s + 2.57e3·29-s + 8.65e3·31-s − 2.38e3·35-s + 1.09e4·37-s + 1.05e3·41-s + 5.90e3·43-s + 3.40e3·45-s + 5.96e3·47-s + 1.20e4·49-s + 2.90e4·53-s − 3.50e3·55-s + 1.39e4·59-s − 3.28e4·61-s − 4.13e4·63-s + 2.36e3·65-s + 6.95e4·67-s + 5.05e4·71-s − 4.67e4·73-s + ⋯ |
| L(s) = 1 | − 0.250·5-s + 1.31·7-s − 9-s + 0.622·11-s − 0.277·13-s + 0.891·17-s + 0.0495·19-s − 0.621·23-s − 0.937·25-s + 0.569·29-s + 1.61·31-s − 0.328·35-s + 1.31·37-s + 0.0975·41-s + 0.486·43-s + 0.250·45-s + 0.393·47-s + 0.719·49-s + 1.42·53-s − 0.156·55-s + 0.520·59-s − 1.13·61-s − 1.31·63-s + 0.0694·65-s + 1.89·67-s + 1.18·71-s − 1.02·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.131516424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131516424\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + 14 T + p^{5} T^{2} \) |
| 7 | \( 1 - 170 T + p^{5} T^{2} \) |
| 11 | \( 1 - 250 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1062 T + p^{5} T^{2} \) |
| 19 | \( 1 - 78 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1576 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2578 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8654 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10986 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1050 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5900 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5962 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13922 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32882 T + p^{5} T^{2} \) |
| 67 | \( 1 - 69566 T + p^{5} T^{2} \) |
| 71 | \( 1 - 50542 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46750 T + p^{5} T^{2} \) |
| 79 | \( 1 - 19348 T + p^{5} T^{2} \) |
| 83 | \( 1 - 87438 T + p^{5} T^{2} \) |
| 89 | \( 1 - 94170 T + p^{5} T^{2} \) |
| 97 | \( 1 - 182786 T + p^{5} T^{2} \) |
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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
−11.78005978336519399690881714594, −10.67564864254429194657924871825, −9.503222991785454417137531095930, −8.291742098767639443296573555133, −7.77462338543806096189055495039, −6.24034028663459677983356653057, −5.15756180336914464906175060893, −3.98860997398042718137541868406, −2.44040998976634732323642933008, −0.927241864813892032394889783112,
0.927241864813892032394889783112, 2.44040998976634732323642933008, 3.98860997398042718137541868406, 5.15756180336914464906175060893, 6.24034028663459677983356653057, 7.77462338543806096189055495039, 8.291742098767639443296573555133, 9.503222991785454417137531095930, 10.67564864254429194657924871825, 11.78005978336519399690881714594
