| L(s) = 1 | + 0.291·2-s + 3-s − 1.91·4-s − 3.62·5-s + 0.291·6-s − 1.35·7-s − 1.14·8-s + 9-s − 1.05·10-s − 11-s − 1.91·12-s − 0.394·14-s − 3.62·15-s + 3.49·16-s − 3.00·17-s + 0.291·18-s + 7.12·19-s + 6.94·20-s − 1.35·21-s − 0.291·22-s − 1.04·23-s − 1.14·24-s + 8.13·25-s + 27-s + 2.59·28-s + 8.54·29-s − 1.05·30-s + ⋯ |
| L(s) = 1 | + 0.206·2-s + 0.577·3-s − 0.957·4-s − 1.62·5-s + 0.119·6-s − 0.511·7-s − 0.403·8-s + 0.333·9-s − 0.334·10-s − 0.301·11-s − 0.552·12-s − 0.105·14-s − 0.935·15-s + 0.874·16-s − 0.728·17-s + 0.0687·18-s + 1.63·19-s + 1.55·20-s − 0.295·21-s − 0.0621·22-s − 0.218·23-s − 0.232·24-s + 1.62·25-s + 0.192·27-s + 0.489·28-s + 1.58·29-s − 0.192·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.291T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 17 | \( 1 + 3.00T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 + 9.77T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 - 9.35T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 + 8.83T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 4.23T + 97T^{2} \) |
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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
−7.961609446873084720599095220169, −7.24318130348096694118167584114, −6.51592064214557526313353949720, −5.38917463205197538120806527198, −4.60757556067559480501727891930, −4.11957937488255235287325727045, −3.27352479871368074096586836465, −2.86234153177309200267659824953, −1.05542423297857918197114995883, 0,
1.05542423297857918197114995883, 2.86234153177309200267659824953, 3.27352479871368074096586836465, 4.11957937488255235287325727045, 4.60757556067559480501727891930, 5.38917463205197538120806527198, 6.51592064214557526313353949720, 7.24318130348096694118167584114, 7.961609446873084720599095220169
