| L(s) = 1 | − 3.37i·3-s − 3.37i·7-s − 8.37·9-s + 11-s − 2i·13-s + 1.37i·17-s + 0.627·19-s − 11.3·21-s + 2.74i·23-s + 18.1i·27-s − 1.37·29-s − 3.37·31-s − 3.37i·33-s + 9.37i·37-s − 6.74·39-s + ⋯ |
| L(s) = 1 | − 1.94i·3-s − 1.27i·7-s − 2.79·9-s + 0.301·11-s − 0.554i·13-s + 0.332i·17-s + 0.144·19-s − 2.48·21-s + 0.572i·23-s + 3.48i·27-s − 0.254·29-s − 0.605·31-s − 0.587i·33-s + 1.54i·37-s − 1.07·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3316294667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3316294667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3.37iT - 3T^{2} \) |
| 7 | \( 1 + 3.37iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 1.37iT - 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 - 2.74iT - 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 - 9.37iT - 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2.74iT - 47T^{2} \) |
| 53 | \( 1 - 4.11iT - 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.4iT - 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + 2.74iT - 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 + 12.7iT - 97T^{2} \) |
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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
−7.52599943485706968751987318481, −7.18445026377063499692972966602, −6.50249783921226254108290353212, −5.82597794915152695227948618028, −4.94914978772049109497872386507, −3.67873383664517524517418738092, −2.98774732300977688835295660072, −1.77020855893307414728841043438, −1.16801658907948672190248690030, −0.093828597756472670784077859717,
2.09322377564879452144978989159, 2.97562575468675274402049678728, 3.70968632322391249084089892410, 4.53178910656260754766161459466, 5.16288920834698826539077202299, 5.77943756141398281180757960966, 6.48282946856966168488423370169, 7.72272039813499957343486580118, 8.747565346826612705162285058200, 8.961704177121559746697378447529
