L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 2·11-s + 13-s + 15-s − 2·17-s − 6·19-s − 21-s − 4·23-s + 25-s − 27-s − 8·29-s + 8·31-s − 2·33-s − 35-s + 2·37-s − 39-s + 6·41-s − 4·43-s − 45-s + 4·47-s + 49-s + 2·51-s + 6·53-s − 2·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s − 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.43·31-s − 0.348·33-s − 0.169·35-s + 0.328·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261781298\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261781298\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.71065944591898, −15.90345722191262, −15.40054158903784, −14.88351217859291, −14.34165779864409, −13.54282142914251, −13.09017280420197, −12.33034994090546, −11.87307881913682, −11.34427668754575, −10.74877843960709, −10.31102872701643, −9.396989406529852, −8.875229380617744, −8.144582273313841, −7.621305739193739, −6.797522874975448, −6.236024917945326, −5.700675189350217, −4.673735766422291, −4.253480122419841, −3.608478220744390, −2.429611489194497, −1.643198372611369, −0.5431821408183072,
0.5431821408183072, 1.643198372611369, 2.429611489194497, 3.608478220744390, 4.253480122419841, 4.673735766422291, 5.700675189350217, 6.236024917945326, 6.797522874975448, 7.621305739193739, 8.144582273313841, 8.875229380617744, 9.396989406529852, 10.31102872701643, 10.74877843960709, 11.34427668754575, 11.87307881913682, 12.33034994090546, 13.09017280420197, 13.54282142914251, 14.34165779864409, 14.88351217859291, 15.40054158903784, 15.90345722191262, 16.71065944591898