| L(s) = 1 | − 2.41·3-s + 5-s + 2.82·9-s − 11-s + 0.414·13-s − 2.41·15-s − 2.41·17-s − 2·19-s + 6.24·23-s + 25-s + 0.414·27-s + 29-s − 10.2·31-s + 2.41·33-s + 11.8·37-s − 0.999·39-s − 4.58·41-s − 11.6·43-s + 2.82·45-s − 7.58·47-s + 5.82·51-s + 6.58·53-s − 55-s + 4.82·57-s − 1.75·59-s + 6.82·61-s + 0.414·65-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.447·5-s + 0.942·9-s − 0.301·11-s + 0.114·13-s − 0.623·15-s − 0.585·17-s − 0.458·19-s + 1.30·23-s + 0.200·25-s + 0.0797·27-s + 0.185·29-s − 1.83·31-s + 0.420·33-s + 1.95·37-s − 0.160·39-s − 0.716·41-s − 1.77·43-s + 0.421·45-s − 1.10·47-s + 0.816·51-s + 0.904·53-s − 0.134·55-s + 0.639·57-s − 0.228·59-s + 0.874·61-s + 0.0513·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 0.414T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 4.58T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 6.58T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 - 6.82T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.894043117173864131026055414015, −7.962828731365968380936562388823, −6.80157668638318403203050778214, −6.50793457503482031950437318612, −5.41787370308164854260839046471, −5.08419882834127589663811770687, −4.00366368575479376304809507031, −2.68699120005286641912367186269, −1.39033773441193893177945600602, 0,
1.39033773441193893177945600602, 2.68699120005286641912367186269, 4.00366368575479376304809507031, 5.08419882834127589663811770687, 5.41787370308164854260839046471, 6.50793457503482031950437318612, 6.80157668638318403203050778214, 7.962828731365968380936562388823, 8.894043117173864131026055414015
