| L(s) = 1 | − 3.73·5-s + 3.46·7-s − 2·11-s + 2.46·13-s + 2.26·17-s + 7.46·19-s − 4.92·23-s + 8.92·25-s − 4.26·29-s − 10.9·31-s − 12.9·35-s + 0.464·37-s + 6.92·41-s + 4.53·43-s + 6.92·47-s + 4.99·49-s + 10.9·53-s + 7.46·55-s − 8·59-s − 10.4·61-s − 9.19·65-s + 0.535·67-s − 2·71-s + 73-s − 6.92·77-s + 0.535·79-s + 2.92·83-s + ⋯ |
| L(s) = 1 | − 1.66·5-s + 1.30·7-s − 0.603·11-s + 0.683·13-s + 0.550·17-s + 1.71·19-s − 1.02·23-s + 1.78·25-s − 0.792·29-s − 1.96·31-s − 2.18·35-s + 0.0762·37-s + 1.08·41-s + 0.691·43-s + 1.01·47-s + 0.714·49-s + 1.50·53-s + 1.00·55-s − 1.04·59-s − 1.33·61-s − 1.14·65-s + 0.0654·67-s − 0.237·71-s + 0.117·73-s − 0.789·77-s + 0.0602·79-s + 0.321·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.540472711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540472711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.73T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 0.464T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 0.535T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.933171613660883810335227684395, −7.60102310079703394996468149925, −7.27749550828776385360926653738, −5.78641717391673061575873287708, −5.32940378436740806065131040417, −4.36767439510945135075757967261, −3.82587537469045610583983175552, −3.05145474673558703570642907281, −1.76432324102927737082926567854, −0.69352687269938136823974124002,
0.69352687269938136823974124002, 1.76432324102927737082926567854, 3.05145474673558703570642907281, 3.82587537469045610583983175552, 4.36767439510945135075757967261, 5.32940378436740806065131040417, 5.78641717391673061575873287708, 7.27749550828776385360926653738, 7.60102310079703394996468149925, 7.933171613660883810335227684395
