| L(s) = 1 | + 2.04·2-s − 3-s + 2.17·4-s − 2.02·5-s − 2.04·6-s − 0.507·7-s + 0.364·8-s + 9-s − 4.13·10-s + 3.20·11-s − 2.17·12-s + 13-s − 1.03·14-s + 2.02·15-s − 3.61·16-s + 0.773·17-s + 2.04·18-s − 2.88·19-s − 4.40·20-s + 0.507·21-s + 6.55·22-s + 4.74·23-s − 0.364·24-s − 0.917·25-s + 2.04·26-s − 27-s − 1.10·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s − 0.577·3-s + 1.08·4-s − 0.903·5-s − 0.834·6-s − 0.191·7-s + 0.129·8-s + 0.333·9-s − 1.30·10-s + 0.966·11-s − 0.628·12-s + 0.277·13-s − 0.277·14-s + 0.521·15-s − 0.902·16-s + 0.187·17-s + 0.481·18-s − 0.661·19-s − 0.984·20-s + 0.110·21-s + 1.39·22-s + 0.988·23-s − 0.0745·24-s − 0.183·25-s + 0.400·26-s − 0.192·27-s − 0.209·28-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 | \( 1 - 2.04T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 0.507T + 7T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 17 | \( 1 - 0.773T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 + 9.06T + 79T^{2} \) |
| 83 | \( 1 - 4.40T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
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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
−7.83492685911843599975526035853, −7.02751989458461328007194279062, −6.40178999524450052986941525911, −5.81438598657924859505756208675, −4.91712451892331198988221417368, −4.26105532883953797214134500879, −3.70221325674691145808544035767, −2.93681069174415722041360356159, −1.55023377929953314173060360192, 0,
1.55023377929953314173060360192, 2.93681069174415722041360356159, 3.70221325674691145808544035767, 4.26105532883953797214134500879, 4.91712451892331198988221417368, 5.81438598657924859505756208675, 6.40178999524450052986941525911, 7.02751989458461328007194279062, 7.83492685911843599975526035853
