| L(s) = 1 | − 2-s − 0.516·3-s + 4-s − 1.53·5-s + 0.516·6-s − 0.576·7-s − 8-s − 2.73·9-s + 1.53·10-s + 11-s − 0.516·12-s + 6.31·13-s + 0.576·14-s + 0.791·15-s + 16-s − 0.519·17-s + 2.73·18-s − 0.276·19-s − 1.53·20-s + 0.298·21-s − 22-s − 1.50·23-s + 0.516·24-s − 2.65·25-s − 6.31·26-s + 2.96·27-s − 0.576·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.298·3-s + 0.5·4-s − 0.685·5-s + 0.210·6-s − 0.218·7-s − 0.353·8-s − 0.910·9-s + 0.484·10-s + 0.301·11-s − 0.149·12-s + 1.75·13-s + 0.154·14-s + 0.204·15-s + 0.250·16-s − 0.126·17-s + 0.644·18-s − 0.0633·19-s − 0.342·20-s + 0.0650·21-s − 0.213·22-s − 0.314·23-s + 0.105·24-s − 0.530·25-s − 1.23·26-s + 0.570·27-s − 0.109·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 0.516T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 + 0.576T + 7T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 0.519T + 17T^{2} \) |
| 19 | \( 1 + 0.276T + 19T^{2} \) |
| 23 | \( 1 + 1.50T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 0.621T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 1.94T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 - 2.60T + 61T^{2} \) |
| 67 | \( 1 - 5.29T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.91T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 + 0.321T + 83T^{2} \) |
| 89 | \( 1 - 7.65T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−8.368929442313135532954453934873, −7.35776730733268803497876979083, −6.54835313136836150553840653105, −5.98721871436547280668913727071, −5.23891409294977528617207451308, −3.90167147193857070540249271579, −3.51375080983995945300633347249, −2.32784074703082020303144573106, −1.13104078781894477721266189341, 0,
1.13104078781894477721266189341, 2.32784074703082020303144573106, 3.51375080983995945300633347249, 3.90167147193857070540249271579, 5.23891409294977528617207451308, 5.98721871436547280668913727071, 6.54835313136836150553840653105, 7.35776730733268803497876979083, 8.368929442313135532954453934873
