L(s) = 1 | + 1.95·2-s + 2.82·4-s − 1.33·5-s + 3.57·8-s − 2.61·10-s + 0.209·11-s + 4.16·16-s + 17-s − 3.78·20-s + 0.408·22-s + 0.790·25-s − 1.82·29-s − 1.95·31-s + 4.57·32-s + 1.95·34-s − 4.78·40-s + 0.591·44-s + 49-s + 1.54·50-s − 0.279·55-s − 3.57·58-s − 61-s − 3.82·62-s + 4.78·64-s − 67-s + 2.82·68-s + 1.61·71-s + ⋯ |
L(s) = 1 | + 1.95·2-s + 2.82·4-s − 1.33·5-s + 3.57·8-s − 2.61·10-s + 0.209·11-s + 4.16·16-s + 17-s − 3.78·20-s + 0.408·22-s + 0.790·25-s − 1.82·29-s − 1.95·31-s + 4.57·32-s + 1.95·34-s − 4.78·40-s + 0.591·44-s + 49-s + 1.54·50-s − 0.279·55-s − 3.57·58-s − 61-s − 3.82·62-s + 4.78·64-s − 67-s + 2.82·68-s + 1.61·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.298158833\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.298158833\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - 1.95T + T^{2} \) |
| 5 | \( 1 + 1.33T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.209T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.82T + T^{2} \) |
| 31 | \( 1 + 1.95T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.82T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.280003542613854084324792374367, −7.970419102302179450553190643691, −7.46956250811155510901380922701, −6.88991608107751183539793903570, −5.76311759166168057555577179649, −5.27925547006209444379858608979, −4.15038502190709643733545701149, −3.77614370213677914234658174189, −3.00731980204421850300805422460, −1.71430248589005186633213733727,
1.71430248589005186633213733727, 3.00731980204421850300805422460, 3.77614370213677914234658174189, 4.15038502190709643733545701149, 5.27925547006209444379858608979, 5.76311759166168057555577179649, 6.88991608107751183539793903570, 7.46956250811155510901380922701, 7.970419102302179450553190643691, 9.280003542613854084324792374367