L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 11-s − 12-s − 13-s + 2·14-s + 16-s − 17-s + 18-s − 2·21-s − 22-s − 4·23-s − 24-s − 26-s − 27-s + 2·28-s + 4·29-s + 2·31-s + 32-s + 33-s − 34-s + 36-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 − 0.301·11-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 − 0.436·21-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.359·31-s + 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.301857587\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.301857587\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.47661289989040, −11.96765586017555, −11.66546777130382, −11.39701334123765, −10.69761328673236, −10.36228867995355, −10.04032152710256, −9.463110438014626, −8.665911643608067, −8.412822920871659, −7.877138290345061, −7.241453515783996, −7.056423320366448, −6.293895110539153, −5.985405806443488, −5.488786557446558, −4.857531099444160, −4.613784251616768, −4.245910158395457, −3.303316332226459, −3.142059865521290, −2.176012942128576, −1.845281829274012, −1.225952698759499, −0.3526128107377795,
0.3526128107377795, 1.225952698759499, 1.845281829274012, 2.176012942128576, 3.142059865521290, 3.303316332226459, 4.245910158395457, 4.613784251616768, 4.857531099444160, 5.488786557446558, 5.985405806443488, 6.293895110539153, 7.056423320366448, 7.241453515783996, 7.877138290345061, 8.412822920871659, 8.665911643608067, 9.463110438014626, 10.04032152710256, 10.36228867995355, 10.69761328673236, 11.39701334123765, 11.66546777130382, 11.96765586017555, 12.47661289989040