| L(s) = 1 | − 2.25·2-s + 127.·3-s − 506.·4-s − 1.28e3·5-s − 287.·6-s + 6.70e3·7-s + 2.29e3·8-s − 3.44e3·9-s + 2.89e3·10-s + 4.29e4·11-s − 6.45e4·12-s − 9.99e4·13-s − 1.51e4·14-s − 1.63e5·15-s + 2.54e5·16-s + 2.09e5·17-s + 7.76e3·18-s + 9.47e5·19-s + 6.50e5·20-s + 8.54e5·21-s − 9.69e4·22-s − 1.96e6·23-s + 2.92e5·24-s − 3.06e5·25-s + 2.25e5·26-s − 2.94e6·27-s − 3.40e6·28-s + ⋯ |
| L(s) = 1 | − 0.0997·2-s + 0.908·3-s − 0.990·4-s − 0.918·5-s − 0.0905·6-s + 1.05·7-s + 0.198·8-s − 0.174·9-s + 0.0915·10-s + 0.885·11-s − 0.899·12-s − 0.970·13-s − 0.105·14-s − 0.833·15-s + 0.970·16-s + 0.608·17-s + 0.0174·18-s + 1.66·19-s + 0.908·20-s + 0.959·21-s − 0.0882·22-s − 1.46·23-s + 0.180·24-s − 0.157·25-s + 0.0967·26-s − 1.06·27-s − 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + 1.41e8T \) |
| good | 2 | \( 1 + 2.25T + 512T^{2} \) |
| 3 | \( 1 - 127.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.28e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.70e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.29e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.99e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.47e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.96e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.20e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.37e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.65e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.51e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.05e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.10e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.55e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.18e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 7.09e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.02e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.22e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.14e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.42e8T + 7.60e17T^{2} \) |
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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
−11.64561743495464997991221109617, −10.00499669529153114472268579754, −9.044334691913153035870062382737, −8.087108816208158052392050679011, −7.54902176584279644246580158861, −5.36704281575750294128704294347, −4.22960776039040236399255069771, −3.25547272308466650697375089612, −1.49211285959204004155745254865, 0,
1.49211285959204004155745254865, 3.25547272308466650697375089612, 4.22960776039040236399255069771, 5.36704281575750294128704294347, 7.54902176584279644246580158861, 8.087108816208158052392050679011, 9.044334691913153035870062382737, 10.00499669529153114472268579754, 11.64561743495464997991221109617
