| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 4·11-s + 13-s − 2·14-s + 16-s + 17-s − 6·19-s + 4·22-s + 4·23-s + 26-s − 2·28-s + 6·29-s − 8·31-s + 32-s + 34-s − 2·37-s − 6·38-s + 6·41-s + 2·43-s + 4·44-s + 4·46-s + 8·47-s − 3·49-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.852·22-s + 0.834·23-s + 0.196·26-s − 0.377·28-s + 1.11·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.973·38-s + 0.937·41-s + 0.304·43-s + 0.603·44-s + 0.589·46-s + 1.16·47-s − 3/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.12744902178913, −13.42123708221859, −13.00325069762432, −12.56257726693657, −12.24770570622930, −11.63044379073516, −11.07188402282035, −10.70047434094023, −10.17866833883534, −9.478469014054034, −9.070085852348465, −8.639005212386561, −7.974699182155248, −7.218783059203020, −6.846818429675954, −6.382313522906080, −5.891290782337477, −5.420534100788424, −4.466678478390138, −4.276138444441217, −3.583060279734088, −3.082457949141129, −2.452606413472380, −1.657639705267797, −1.032454947044504, 0,
1.032454947044504, 1.657639705267797, 2.452606413472380, 3.082457949141129, 3.583060279734088, 4.276138444441217, 4.466678478390138, 5.420534100788424, 5.891290782337477, 6.382313522906080, 6.846818429675954, 7.218783059203020, 7.974699182155248, 8.639005212386561, 9.070085852348465, 9.478469014054034, 10.17866833883534, 10.70047434094023, 11.07188402282035, 11.63044379073516, 12.24770570622930, 12.56257726693657, 13.00325069762432, 13.42123708221859, 14.12744902178913
