| L(s) = 1 | + 2.14·2-s + 1.43·3-s + 2.58·4-s + 3.06·6-s + 0.0457·7-s + 1.25·8-s − 0.953·9-s − 4.86·11-s + 3.70·12-s − 5.72·13-s + 0.0978·14-s − 2.48·16-s + 5.91·17-s − 2.04·18-s + 7.44·19-s + 0.0653·21-s − 10.4·22-s − 6.30·23-s + 1.79·24-s − 12.2·26-s − 5.65·27-s + 0.118·28-s − 8.13·29-s − 9.12·31-s − 7.82·32-s − 6.96·33-s + 12.6·34-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.826·3-s + 1.29·4-s + 1.25·6-s + 0.0172·7-s + 0.444·8-s − 0.317·9-s − 1.46·11-s + 1.06·12-s − 1.58·13-s + 0.0261·14-s − 0.620·16-s + 1.43·17-s − 0.481·18-s + 1.70·19-s + 0.0142·21-s − 2.22·22-s − 1.31·23-s + 0.366·24-s − 2.40·26-s − 1.08·27-s + 0.0223·28-s − 1.51·29-s − 1.63·31-s − 1.38·32-s − 1.21·33-s + 2.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 - 1.43T + 3T^{2} \) |
| 7 | \( 1 - 0.0457T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 - 5.91T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 6.19T + 41T^{2} \) |
| 43 | \( 1 - 3.16T + 43T^{2} \) |
| 47 | \( 1 - 0.163T + 47T^{2} \) |
| 53 | \( 1 - 8.93T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 + 5.15T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 + 5.93T + 73T^{2} \) |
| 79 | \( 1 - 5.05T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.60140671104030656559556460111, −7.21875072413728433576160505232, −5.73887231172169301022291586825, −5.52580563975945922550652388492, −4.98871646303339975579360266111, −3.86033935720098303419219871807, −3.30126408530280104005984334948, −2.63862017745108113295227676687, −2.00690884867560741552483267438, 0,
2.00690884867560741552483267438, 2.63862017745108113295227676687, 3.30126408530280104005984334948, 3.86033935720098303419219871807, 4.98871646303339975579360266111, 5.52580563975945922550652388492, 5.73887231172169301022291586825, 7.21875072413728433576160505232, 7.60140671104030656559556460111
