L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 2·13-s − 14-s + 16-s + 20-s + 23-s + 25-s − 2·26-s − 28-s − 8·29-s − 4·31-s + 32-s − 35-s + 6·37-s + 40-s − 12·43-s + 46-s + 4·47-s + 49-s + 50-s − 2·52-s − 4·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.48·29-s − 0.718·31-s + 0.176·32-s − 0.169·35-s + 0.986·37-s + 0.158·40-s − 1.82·43-s + 0.147·46-s + 0.583·47-s + 1/7·49-s + 0.141·50-s − 0.277·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.44425189091883, −15.69346967231024, −15.19815849793927, −14.59590421619927, −14.26944537445959, −13.43930238582343, −13.07721875125000, −12.63888394346532, −11.95833400818978, −11.32593604530458, −10.86085865841818, −10.07184819668084, −9.603343223511207, −9.055692106085130, −8.176565641841324, −7.554168209674948, −6.847663876460818, −6.420015073659919, −5.460688049658179, −5.306180728740109, −4.303547598383556, −3.692733318125969, −2.901102417223302, −2.210375369926814, −1.375346648774675, 0,
1.375346648774675, 2.210375369926814, 2.901102417223302, 3.692733318125969, 4.303547598383556, 5.306180728740109, 5.460688049658179, 6.420015073659919, 6.847663876460818, 7.554168209674948, 8.176565641841324, 9.055692106085130, 9.603343223511207, 10.07184819668084, 10.86085865841818, 11.32593604530458, 11.95833400818978, 12.63888394346532, 13.07721875125000, 13.43930238582343, 14.26944537445959, 14.59590421619927, 15.19815849793927, 15.69346967231024, 16.44425189091883