| L(s) = 1 | + 3·3-s − 5-s + 6·9-s + 5·11-s − 3·13-s − 3·15-s + 17-s + 6·19-s + 6·23-s + 25-s + 9·27-s + 9·29-s + 4·31-s + 15·33-s − 2·37-s − 9·39-s + 4·41-s − 10·43-s − 6·45-s + 47-s + 3·51-s − 4·53-s − 5·55-s + 18·57-s − 8·59-s − 8·61-s + 3·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s − 0.832·13-s − 0.774·15-s + 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s + 0.718·31-s + 2.61·33-s − 0.328·37-s − 1.44·39-s + 0.624·41-s − 1.52·43-s − 0.894·45-s + 0.145·47-s + 0.420·51-s − 0.549·53-s − 0.674·55-s + 2.38·57-s − 1.04·59-s − 1.02·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.177484264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.177484264\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.68637257465096, −15.34881960063125, −14.83287848882044, −14.22620432200799, −14.00948196313391, −13.45136754007393, −12.67787903545574, −12.04132386571269, −11.78617901180500, −10.85348054301284, −10.02436093194460, −9.625445348722671, −9.027594370038392, −8.680983889976861, −7.894064411075884, −7.481372508201108, −6.860876659378844, −6.298090790647029, −4.984850858359601, −4.579814849572540, −3.739356789545451, −3.128868594840071, −2.754564713556051, −1.618837396431284, −0.9877724393290895,
0.9877724393290895, 1.618837396431284, 2.754564713556051, 3.128868594840071, 3.739356789545451, 4.579814849572540, 4.984850858359601, 6.298090790647029, 6.860876659378844, 7.481372508201108, 7.894064411075884, 8.680983889976861, 9.027594370038392, 9.625445348722671, 10.02436093194460, 10.85348054301284, 11.78617901180500, 12.04132386571269, 12.67787903545574, 13.45136754007393, 14.00948196313391, 14.22620432200799, 14.83287848882044, 15.34881960063125, 15.68637257465096
