| L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 8-s − 2·9-s + 4·10-s + 2·11-s + 12-s + 6·13-s + 4·15-s + 16-s − 3·17-s − 2·18-s + 4·20-s + 2·22-s − 6·23-s + 24-s + 11·25-s + 6·26-s − 5·27-s + 4·30-s + 3·31-s + 32-s + 2·33-s − 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.288·12-s + 1.66·13-s + 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.894·20-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 1.17·26-s − 0.962·27-s + 0.730·30-s + 0.538·31-s + 0.176·32-s + 0.348·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.948713870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.948713870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.92502186661398, −15.45696510623977, −14.64738531997057, −14.33902734096606, −13.67802240622094, −13.39920977392474, −13.25271210989489, −12.12391620733696, −11.68830948246823, −10.96477871418647, −10.36392363406559, −9.851345215885561, −9.140966131542003, −8.533350533440568, −8.307203157193371, −6.999186035082532, −6.498090827403802, −5.979074153753644, −5.558577075487731, −4.788966223082288, −3.756269898774884, −3.389644181468793, −2.222757130616284, −2.070863032966492, −1.042725259280208,
1.042725259280208, 2.070863032966492, 2.222757130616284, 3.389644181468793, 3.756269898774884, 4.788966223082288, 5.558577075487731, 5.979074153753644, 6.498090827403802, 6.999186035082532, 8.307203157193371, 8.533350533440568, 9.140966131542003, 9.851345215885561, 10.36392363406559, 10.96477871418647, 11.68830948246823, 12.12391620733696, 13.25271210989489, 13.39920977392474, 13.67802240622094, 14.33902734096606, 14.64738531997057, 15.45696510623977, 15.92502186661398
