| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 4·7-s − 8-s + 9-s + 2·12-s − 2·13-s + 4·14-s + 16-s − 18-s + 4·19-s − 8·21-s − 2·24-s − 5·25-s + 2·26-s − 4·27-s − 4·28-s + 4·31-s − 32-s + 36-s + 4·37-s − 4·38-s − 4·39-s + 6·41-s + 8·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 1.74·21-s − 0.408·24-s − 25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.648·38-s − 0.640·39-s + 0.937·41-s + 1.23·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.090575850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090575850\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.07855980547436, −13.63252430021753, −13.30033540878259, −12.62891087751724, −12.18849610724332, −11.66128070237924, −11.03415964809628, −10.39606810893874, −9.650049428100523, −9.564565059072523, −9.360407013456906, −8.465227519469344, −8.029277726046875, −7.671268819359817, −6.899102043090937, −6.619942541288945, −5.841198185246937, −5.391287391175485, −4.365531345353742, −3.751006955506940, −3.122518749939049, −2.793879254807793, −2.197120704406993, −1.343115406888483, −0.3575279039990130,
0.3575279039990130, 1.343115406888483, 2.197120704406993, 2.793879254807793, 3.122518749939049, 3.751006955506940, 4.365531345353742, 5.391287391175485, 5.841198185246937, 6.619942541288945, 6.899102043090937, 7.671268819359817, 8.029277726046875, 8.465227519469344, 9.360407013456906, 9.564565059072523, 9.650049428100523, 10.39606810893874, 11.03415964809628, 11.66128070237924, 12.18849610724332, 12.62891087751724, 13.30033540878259, 13.63252430021753, 14.07855980547436
