| L(s) = 1 | − 4.31·3-s + 44.7·5-s − 224.·9-s − 435.·11-s − 455.·13-s − 193.·15-s + 1.01e3·17-s + 594.·19-s + 3.03e3·23-s − 1.12e3·25-s + 2.01e3·27-s + 3.96e3·29-s + 772.·31-s + 1.88e3·33-s + 3.19e3·37-s + 1.96e3·39-s − 1.33e4·41-s − 6.16e3·43-s − 1.00e4·45-s + 1.41e3·47-s − 4.37e3·51-s + 1.43e4·53-s − 1.94e4·55-s − 2.56e3·57-s + 4.28e4·59-s − 1.67e4·61-s − 2.03e4·65-s + ⋯ |
| L(s) = 1 | − 0.277·3-s + 0.800·5-s − 0.923·9-s − 1.08·11-s − 0.746·13-s − 0.221·15-s + 0.850·17-s + 0.377·19-s + 1.19·23-s − 0.359·25-s + 0.532·27-s + 0.874·29-s + 0.144·31-s + 0.300·33-s + 0.383·37-s + 0.206·39-s − 1.24·41-s − 0.508·43-s − 0.738·45-s + 0.0936·47-s − 0.235·51-s + 0.700·53-s − 0.868·55-s − 0.104·57-s + 1.60·59-s − 0.577·61-s − 0.597·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.720987346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720987346\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4.31T + 243T^{2} \) |
| 5 | \( 1 - 44.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 435.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 455.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 594.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 772.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.16e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.15e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.38e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.14e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.35e5T + 8.58e9T^{2} \) |
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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
−10.35404866904728620051479319370, −9.772714080504751764851407360703, −8.655211245893896316698347252448, −7.72774670728290623949943068388, −6.60371377675636854705806993442, −5.48437431111829969662387514957, −5.03488568564410582489446865179, −3.19075464229673658739893255128, −2.27381034965173137995975155337, −0.68925844704766466453520335988,
0.68925844704766466453520335988, 2.27381034965173137995975155337, 3.19075464229673658739893255128, 5.03488568564410582489446865179, 5.48437431111829969662387514957, 6.60371377675636854705806993442, 7.72774670728290623949943068388, 8.655211245893896316698347252448, 9.772714080504751764851407360703, 10.35404866904728620051479319370
