| L(s) = 1 | + 2·3-s − 7-s + 9-s − 2·11-s − 6·13-s − 6·17-s − 2·21-s + 23-s − 4·27-s + 2·29-s − 8·31-s − 4·33-s + 2·37-s − 12·39-s − 2·41-s + 4·43-s + 6·47-s + 49-s − 12·51-s − 14·53-s + 12·59-s − 63-s − 12·67-s + 2·69-s + 12·71-s − 6·73-s + 2·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 1.45·17-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 1.92·39-s − 0.312·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 1.68·51-s − 1.92·53-s + 1.56·59-s − 0.125·63-s − 1.46·67-s + 0.240·69-s + 1.42·71-s − 0.702·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7694169050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7694169050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.23132197731012, −13.85924577682844, −13.08995740725573, −12.95512446635738, −12.41142807351176, −11.72500490963125, −11.15365110857884, −10.65006668147077, −9.995232006705254, −9.542796433387516, −9.109386948962753, −8.658777916567256, −8.050246577332205, −7.524974095724326, −7.081473117243959, −6.594036023228363, −5.668645095907518, −5.271097743743150, −4.416723112364775, −4.110448915701149, −3.154472474286590, −2.719049106613662, −2.298600372126727, −1.621969659927204, −0.2494477513874447,
0.2494477513874447, 1.621969659927204, 2.298600372126727, 2.719049106613662, 3.154472474286590, 4.110448915701149, 4.416723112364775, 5.271097743743150, 5.668645095907518, 6.594036023228363, 7.081473117243959, 7.524974095724326, 8.050246577332205, 8.658777916567256, 9.109386948962753, 9.542796433387516, 9.995232006705254, 10.65006668147077, 11.15365110857884, 11.72500490963125, 12.41142807351176, 12.95512446635738, 13.08995740725573, 13.85924577682844, 14.23132197731012
