L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s + 15-s + 6·17-s − 4·19-s + 21-s + 4·23-s + 25-s − 27-s − 10·29-s + 8·31-s − 4·33-s + 35-s + 6·37-s + 39-s − 2·41-s − 4·43-s − 45-s + 49-s − 6·51-s + 2·53-s − 4·55-s + 4·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s + 0.169·35-s + 0.986·37-s + 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 0.529·57-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
\(L(1)\) |
\(\approx\) |
\(1.519068713\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.519068713\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.71070955647353, −16.06441246869183, −15.27183055736836, −14.88900709663561, −14.37158180071427, −13.63081864279290, −12.82797375104831, −12.54108186812937, −11.79010365411597, −11.44052125086744, −10.81809166184903, −9.984876608300101, −9.620918419544967, −8.906336367459577, −8.146421129921662, −7.525465333418808, −6.773460423998573, −6.365374895976331, −5.562291817291877, −4.916968046941218, −4.011502495628750, −3.605850222744458, −2.612638672442301, −1.473785896898834, −0.6318765988230696,
0.6318765988230696, 1.473785896898834, 2.612638672442301, 3.605850222744458, 4.011502495628750, 4.916968046941218, 5.562291817291877, 6.365374895976331, 6.773460423998573, 7.525465333418808, 8.146421129921662, 8.906336367459577, 9.620918419544967, 9.984876608300101, 10.81809166184903, 11.44052125086744, 11.79010365411597, 12.54108186812937, 12.82797375104831, 13.63081864279290, 14.37158180071427, 14.88900709663561, 15.27183055736836, 16.06441246869183, 16.71070955647353