| L(s) = 1 | − 1.41·2-s + 3-s + 3.41·5-s − 1.41·6-s − 3.41·7-s + 2.82·8-s + 9-s − 4.82·10-s + 2.41·11-s + 6.24·13-s + 4.82·14-s + 3.41·15-s − 4.00·16-s − 0.414·17-s − 1.41·18-s − 5.41·19-s − 3.41·21-s − 3.41·22-s − 1.41·23-s + 2.82·24-s + 6.65·25-s − 8.82·26-s + 27-s − 6.07·29-s − 4.82·30-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.577·3-s + 1.52·5-s − 0.577·6-s − 1.29·7-s + 0.999·8-s + 0.333·9-s − 1.52·10-s + 0.727·11-s + 1.73·13-s + 1.29·14-s + 0.881·15-s − 1.00·16-s − 0.100·17-s − 0.333·18-s − 1.24·19-s − 0.745·21-s − 0.727·22-s − 0.294·23-s + 0.577·24-s + 1.33·25-s − 1.73·26-s + 0.192·27-s − 1.12·29-s − 0.881·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8548655089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8548655089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 0.414T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 6.07T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 9.48T + 37T^{2} \) |
| 43 | \( 1 + 5T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 6.82T + 53T^{2} \) |
| 59 | \( 1 + 8.48T + 59T^{2} \) |
| 61 | \( 1 + 4.65T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 7.65T + 79T^{2} \) |
| 83 | \( 1 - 1.07T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−13.46399897906393686856027095324, −12.81844717494374397842697575262, −10.78347769354339306901080134615, −9.994795573797464625637559537776, −9.121296601361451419041207696796, −8.663540701558035472144006964402, −6.86136765449303973440793018188, −5.92263008191291302094574962741, −3.76990265978882257353141619058, −1.77891449336676596242213201754,
1.77891449336676596242213201754, 3.76990265978882257353141619058, 5.92263008191291302094574962741, 6.86136765449303973440793018188, 8.663540701558035472144006964402, 9.121296601361451419041207696796, 9.994795573797464625637559537776, 10.78347769354339306901080134615, 12.81844717494374397842697575262, 13.46399897906393686856027095324
