| L(s) = 1 | − 2·4-s − 4·5-s + 2·7-s + 11-s − 13-s + 4·16-s + 3·17-s + 8·20-s − 2·23-s + 11·25-s − 4·28-s − 4·29-s + 4·31-s − 8·35-s − 2·37-s + 6·41-s − 4·43-s − 2·44-s − 6·47-s − 3·49-s + 2·52-s + 53-s − 4·55-s + 11·59-s + 10·61-s − 8·64-s + 4·65-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 16-s + 0.727·17-s + 1.78·20-s − 0.417·23-s + 11/5·25-s − 0.755·28-s − 0.742·29-s + 0.718·31-s − 1.35·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.301·44-s − 0.875·47-s − 3/7·49-s + 0.277·52-s + 0.137·53-s − 0.539·55-s + 1.43·59-s + 1.28·61-s − 64-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098227910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098227910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.85253344472020, −14.51699988174757, −14.04199292892325, −13.30656503699264, −12.73616579760375, −12.23221878301630, −11.81991870338001, −11.29570395596966, −10.87997438690347, −9.969567940192457, −9.709638937823295, −8.732600802967118, −8.466902313286008, −7.933565047901953, −7.551996138111253, −6.951072105272547, −6.083526950385420, −5.246969638857433, −4.808304502939040, −4.293300189764415, −3.616220253421320, −3.370285188638210, −2.200146320110244, −1.109636628735825, −0.4760028863335252,
0.4760028863335252, 1.109636628735825, 2.200146320110244, 3.370285188638210, 3.616220253421320, 4.293300189764415, 4.808304502939040, 5.246969638857433, 6.083526950385420, 6.951072105272547, 7.551996138111253, 7.933565047901953, 8.466902313286008, 8.732600802967118, 9.709638937823295, 9.969567940192457, 10.87997438690347, 11.29570395596966, 11.81991870338001, 12.23221878301630, 12.73616579760375, 13.30656503699264, 14.04199292892325, 14.51699988174757, 14.85253344472020
