| L(s) = 1 | + 0.602·3-s + 3.63·7-s − 2.63·9-s − 2·11-s − 13-s + 6.67·17-s − 8.06·19-s + 2.19·21-s + 0.794·23-s − 3.39·27-s − 8.06·29-s − 5.20·31-s − 1.20·33-s − 0.431·37-s − 0.602·39-s + 6.86·41-s − 5.80·43-s − 7.63·47-s + 6.22·49-s + 4.02·51-s − 0.794·53-s − 4.86·57-s − 8.06·59-s − 2.86·61-s − 9.58·63-s − 5.20·67-s + 0.478·69-s + ⋯ |
| L(s) = 1 | + 0.347·3-s + 1.37·7-s − 0.878·9-s − 0.603·11-s − 0.277·13-s + 1.61·17-s − 1.85·19-s + 0.478·21-s + 0.165·23-s − 0.653·27-s − 1.49·29-s − 0.934·31-s − 0.209·33-s − 0.0709·37-s − 0.0965·39-s + 1.07·41-s − 0.885·43-s − 1.11·47-s + 0.889·49-s + 0.562·51-s − 0.109·53-s − 0.644·57-s − 1.05·59-s − 0.366·61-s − 1.20·63-s − 0.635·67-s + 0.0576·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.602T + 3T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 6.67T + 17T^{2} \) |
| 19 | \( 1 + 8.06T + 19T^{2} \) |
| 23 | \( 1 - 0.794T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 + 0.431T + 37T^{2} \) |
| 41 | \( 1 - 6.86T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 + 0.794T + 53T^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 + 5.20T + 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 2.06T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.900350470260432016681187262233, −7.47693838971311748416533346215, −6.32192704665946779294617477046, −5.50477744897944542509510436472, −5.06642770528810353965784670029, −4.11067783104357700695355374461, −3.25233306211613965800795414557, −2.29291845786950387715441244510, −1.57438990466819598877145089445, 0,
1.57438990466819598877145089445, 2.29291845786950387715441244510, 3.25233306211613965800795414557, 4.11067783104357700695355374461, 5.06642770528810353965784670029, 5.50477744897944542509510436472, 6.32192704665946779294617477046, 7.47693838971311748416533346215, 7.900350470260432016681187262233
