| L(s) = 1 | + 2-s − 4-s − 3·8-s + 2·11-s + 6·13-s − 16-s + 6·17-s + 6·19-s + 2·22-s − 4·23-s + 6·26-s − 8·29-s + 6·31-s + 5·32-s + 6·34-s + 6·37-s + 6·38-s − 6·41-s − 2·44-s − 4·46-s − 6·52-s + 2·53-s − 8·58-s − 12·59-s + 6·62-s + 7·64-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.603·11-s + 1.66·13-s − 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.426·22-s − 0.834·23-s + 1.17·26-s − 1.48·29-s + 1.07·31-s + 0.883·32-s + 1.02·34-s + 0.986·37-s + 0.973·38-s − 0.937·41-s − 0.301·44-s − 0.589·46-s − 0.832·52-s + 0.274·53-s − 1.05·58-s − 1.56·59-s + 0.762·62-s + 7/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.055775658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.055775658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.46013016401643, −15.87689909392640, −15.23063491818718, −14.69468231815389, −14.07692354919444, −13.60537236389792, −13.35373713450540, −12.43216857880638, −11.99213870197887, −11.52250586377623, −10.79139415353811, −9.944014044996831, −9.459889906081372, −8.949777106087611, −8.061708648427109, −7.775910130823162, −6.646171285232532, −6.020351097568707, −5.591082529048530, −4.886522192828421, −3.927730949920945, −3.600306650973034, −2.944056965893721, −1.559003025695040, −0.7959091287441254,
0.7959091287441254, 1.559003025695040, 2.944056965893721, 3.600306650973034, 3.927730949920945, 4.886522192828421, 5.591082529048530, 6.020351097568707, 6.646171285232532, 7.775910130823162, 8.061708648427109, 8.949777106087611, 9.459889906081372, 9.944014044996831, 10.79139415353811, 11.52250586377623, 11.99213870197887, 12.43216857880638, 13.35373713450540, 13.60537236389792, 14.07692354919444, 14.69468231815389, 15.23063491818718, 15.87689909392640, 16.46013016401643
