| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s + 9-s + 12-s + 13-s − 2·14-s + 16-s + 17-s − 18-s + 2·19-s + 2·21-s + 8·23-s − 24-s − 26-s + 27-s + 2·28-s + 6·29-s − 4·31-s − 32-s − 34-s + 36-s − 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.436·21-s + 1.66·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.828540490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.828540490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.11223339823400, −14.53010253356402, −13.99933974084257, −13.62793971549646, −12.73497561737695, −12.48352279731673, −11.69038496505571, −11.18800957857490, −10.70084251997898, −10.26980478893559, −9.301605342825601, −9.242049980951976, −8.558653675968442, −7.921207713949228, −7.621795156565208, −6.928366573773286, −6.386392046559594, −5.554783536375881, −4.934630766441888, −4.343971241718698, −3.381364247655272, −2.952399163388624, −2.111305942861250, −1.382428584207754, −0.7448741571945212,
0.7448741571945212, 1.382428584207754, 2.111305942861250, 2.952399163388624, 3.381364247655272, 4.343971241718698, 4.934630766441888, 5.554783536375881, 6.386392046559594, 6.928366573773286, 7.621795156565208, 7.921207713949228, 8.558653675968442, 9.242049980951976, 9.301605342825601, 10.26980478893559, 10.70084251997898, 11.18800957857490, 11.69038496505571, 12.48352279731673, 12.73497561737695, 13.62793971549646, 13.99933974084257, 14.53010253356402, 15.11223339823400
