| L(s) = 1 | − 3-s + 5-s − 3.50·7-s + 9-s − 5.96·11-s + 13-s − 15-s − 1.50·17-s − 7.78·19-s + 3.50·21-s − 3.50·23-s + 25-s − 27-s − 5.78·29-s + 5.96·33-s − 3.50·35-s − 3.96·37-s − 39-s + 7.96·41-s − 1.68·43-s + 45-s + 5.27·49-s + 1.50·51-s − 5.65·53-s − 5.96·55-s + 7.78·57-s + 14.9·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.32·7-s + 0.333·9-s − 1.79·11-s + 0.277·13-s − 0.258·15-s − 0.364·17-s − 1.78·19-s + 0.764·21-s − 0.730·23-s + 0.200·25-s − 0.192·27-s − 1.07·29-s + 1.03·33-s − 0.592·35-s − 0.652·37-s − 0.160·39-s + 1.24·41-s − 0.257·43-s + 0.149·45-s + 0.754·49-s + 0.210·51-s − 0.777·53-s − 0.804·55-s + 1.03·57-s + 1.94·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4604112943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4604112943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.50T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 7.78T + 19T^{2} \) |
| 23 | \( 1 + 3.50T + 23T^{2} \) |
| 29 | \( 1 + 5.78T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 + 0.648T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 5.28T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−8.077637504711986877471067603561, −7.15961734163024150288555720830, −6.55259566985638289916669195207, −5.88332275282439837483119550641, −5.42574401447106200765041323142, −4.45300538155240881407442659138, −3.65657849997430554251799855722, −2.64380524632807476617578407790, −2.00692319514159609068048610709, −0.33729451702075681619960265838,
0.33729451702075681619960265838, 2.00692319514159609068048610709, 2.64380524632807476617578407790, 3.65657849997430554251799855722, 4.45300538155240881407442659138, 5.42574401447106200765041323142, 5.88332275282439837483119550641, 6.55259566985638289916669195207, 7.15961734163024150288555720830, 8.077637504711986877471067603561
