| L(s) = 1 | − 2.61·3-s − 1.61·5-s − 3.85·7-s + 3.85·9-s + 2.38·13-s + 4.23·15-s + 2.38·17-s + 3.85·19-s + 10.0·21-s − 2.47·23-s − 2.38·25-s − 2.23·27-s + 8.61·29-s + 0.854·31-s + 6.23·35-s − 1.85·37-s − 6.23·39-s + 8.61·41-s − 6.23·45-s + 1.38·47-s + 7.85·49-s − 6.23·51-s − 4.09·53-s − 10.0·57-s + 1.09·59-s + 2.38·61-s − 14.8·63-s + ⋯ |
| L(s) = 1 | − 1.51·3-s − 0.723·5-s − 1.45·7-s + 1.28·9-s + 0.660·13-s + 1.09·15-s + 0.577·17-s + 0.884·19-s + 2.20·21-s − 0.515·23-s − 0.476·25-s − 0.430·27-s + 1.60·29-s + 0.153·31-s + 1.05·35-s − 0.304·37-s − 0.998·39-s + 1.34·41-s − 0.929·45-s + 0.201·47-s + 1.12·49-s − 0.873·51-s − 0.561·53-s − 1.33·57-s + 0.141·59-s + 0.304·61-s − 1.87·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5461028114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5461028114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 0.854T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 1.38T + 47T^{2} \) |
| 53 | \( 1 + 4.09T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.909T + 73T^{2} \) |
| 79 | \( 1 + 7.14T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−11.06430110269929125618086612169, −10.22125607828448056694602331944, −9.499091209846609372816113826745, −8.143151833804893376361060317778, −7.03403057861190055180506212572, −6.25842546082889903858342915431, −5.54440802512060329956946081750, −4.26340714178821576127283581923, −3.19469397959862183194550061296, −0.71976948220384209893392836349,
0.71976948220384209893392836349, 3.19469397959862183194550061296, 4.26340714178821576127283581923, 5.54440802512060329956946081750, 6.25842546082889903858342915431, 7.03403057861190055180506212572, 8.143151833804893376361060317778, 9.499091209846609372816113826745, 10.22125607828448056694602331944, 11.06430110269929125618086612169
