| L(s) = 1 | − 3.06·3-s − 2.61·5-s + 6.40·9-s − 3.48·11-s − 0.0840·13-s + 8.01·15-s + 7.37·17-s − 6.02·19-s − 6.92·23-s + 1.82·25-s − 10.4·27-s + 29-s + 4.91·31-s + 10.6·33-s − 6.66·37-s + 0.257·39-s + 8.54·41-s + 5.19·43-s − 16.7·45-s − 11.7·47-s − 22.6·51-s + 1.28·53-s + 9.09·55-s + 18.4·57-s + 0.405·59-s − 2.29·61-s + 0.219·65-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 1.16·5-s + 2.13·9-s − 1.05·11-s − 0.0233·13-s + 2.06·15-s + 1.78·17-s − 1.38·19-s − 1.44·23-s + 0.364·25-s − 2.00·27-s + 0.185·29-s + 0.883·31-s + 1.85·33-s − 1.09·37-s + 0.0412·39-s + 1.33·41-s + 0.791·43-s − 2.49·45-s − 1.71·47-s − 3.16·51-s + 0.175·53-s + 1.22·55-s + 2.44·57-s + 0.0527·59-s − 0.293·61-s + 0.0272·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2517440139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2517440139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 3.48T + 11T^{2} \) |
| 13 | \( 1 + 0.0840T + 13T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 + 6.02T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 31 | \( 1 - 4.91T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 - 8.54T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 1.28T + 53T^{2} \) |
| 59 | \( 1 - 0.405T + 59T^{2} \) |
| 61 | \( 1 + 2.29T + 61T^{2} \) |
| 67 | \( 1 + 6.42T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 1.75T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 + 7.10T + 97T^{2} \) |
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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
−7.81181868216785234920905725369, −7.53465395351468326523729499332, −6.56288507721872429902290177168, −5.89072359223469299972163006176, −5.37434334733174155830155880925, −4.46920731535014445054079230956, −4.04741977380802803231693039057, −2.91479731639571075670602285605, −1.50172261195184875254272990678, −0.30185529611633665591957374211,
0.30185529611633665591957374211, 1.50172261195184875254272990678, 2.91479731639571075670602285605, 4.04741977380802803231693039057, 4.46920731535014445054079230956, 5.37434334733174155830155880925, 5.89072359223469299972163006176, 6.56288507721872429902290177168, 7.53465395351468326523729499332, 7.81181868216785234920905725369
