| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·13-s − 15-s + 4·17-s + 19-s − 2·21-s + 4·23-s + 25-s − 27-s − 6·29-s + 10·31-s + 2·35-s − 10·37-s + 2·39-s + 4·43-s + 45-s + 4·47-s − 3·49-s − 4·51-s + 10·53-s − 57-s − 6·59-s + 10·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.970·17-s + 0.229·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.338·35-s − 1.64·37-s + 0.320·39-s + 0.609·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.560·51-s + 1.37·53-s − 0.132·57-s − 0.781·59-s + 1.28·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797748963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797748963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.069081413915831374961589464151, −8.196895120281225132319159157381, −7.40639175626761745320894479931, −6.72847315290697703536252833157, −5.66754873741922586891610116007, −5.20364571383467454845376550496, −4.36995169752943436765696789071, −3.19396417717054129763552092617, −2.02714238275193470939260564344, −0.936846893772393691857917949059,
0.936846893772393691857917949059, 2.02714238275193470939260564344, 3.19396417717054129763552092617, 4.36995169752943436765696789071, 5.20364571383467454845376550496, 5.66754873741922586891610116007, 6.72847315290697703536252833157, 7.40639175626761745320894479931, 8.196895120281225132319159157381, 9.069081413915831374961589464151
