| L(s) = 1 | + 1.46·2-s − 3-s + 0.151·4-s + 0.636·5-s − 1.46·6-s − 0.403·7-s − 2.71·8-s + 9-s + 0.933·10-s − 2.09·11-s − 0.151·12-s − 13-s − 0.591·14-s − 0.636·15-s − 4.28·16-s + 4.83·17-s + 1.46·18-s + 6.09·19-s + 0.0966·20-s + 0.403·21-s − 3.07·22-s + 5.96·23-s + 2.71·24-s − 4.59·25-s − 1.46·26-s − 27-s − 0.0612·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0759·4-s + 0.284·5-s − 0.598·6-s − 0.152·7-s − 0.958·8-s + 0.333·9-s + 0.295·10-s − 0.632·11-s − 0.0438·12-s − 0.277·13-s − 0.158·14-s − 0.164·15-s − 1.07·16-s + 1.17·17-s + 0.345·18-s + 1.39·19-s + 0.0216·20-s + 0.0880·21-s − 0.656·22-s + 1.24·23-s + 0.553·24-s − 0.919·25-s − 0.287·26-s − 0.192·27-s − 0.0115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 5 | \( 1 - 0.636T + 5T^{2} \) |
| 7 | \( 1 + 0.403T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 - 6.09T + 19T^{2} \) |
| 23 | \( 1 - 5.96T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 + 6.81T + 31T^{2} \) |
| 37 | \( 1 + 0.367T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 - 0.519T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 3.66T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 - 0.651T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.86T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−7.69887859604638067965730996853, −7.38597078133199014700901205528, −6.24124116046096255494515296759, −5.56702118464879010910423100092, −5.27635735993423818795891490853, −4.42330002856931673858717970587, −3.43624388980293286908118417453, −2.86908546657632077088358629535, −1.46068940962612500060502169317, 0,
1.46068940962612500060502169317, 2.86908546657632077088358629535, 3.43624388980293286908118417453, 4.42330002856931673858717970587, 5.27635735993423818795891490853, 5.56702118464879010910423100092, 6.24124116046096255494515296759, 7.38597078133199014700901205528, 7.69887859604638067965730996853
