| L(s) = 1 | + 1.24·5-s + 9-s + 1.80·11-s − 1.80·17-s + 0.445·19-s + 0.554·25-s − 0.445·29-s − 1.24·31-s − 1.80·37-s − 1.24·43-s + 1.24·45-s + 0.445·47-s + 49-s + 2.24·55-s − 1.24·59-s + 81-s − 2.24·85-s + 0.554·95-s − 0.445·97-s + 1.80·99-s + 1.80·103-s + ⋯ |
| L(s) = 1 | + 1.24·5-s + 9-s + 1.80·11-s − 1.80·17-s + 0.445·19-s + 0.554·25-s − 0.445·29-s − 1.24·31-s − 1.80·37-s − 1.24·43-s + 1.24·45-s + 0.445·47-s + 49-s + 2.24·55-s − 1.24·59-s + 81-s − 2.24·85-s + 0.554·95-s − 0.445·97-s + 1.80·99-s + 1.80·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.673495073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673495073\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 151 | \( 1 + T \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.80T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 - 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 + 1.24T + T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.24T + T^{2} \) |
| 47 | \( 1 - 0.445T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.24T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−9.108886333357813900499722783684, −8.789263205203508995070140004837, −7.32528635712878165456751149045, −6.76172135011007272509656671776, −6.19873567972955490648429131697, −5.23563368368958247950216116249, −4.32007388267191401416371115554, −3.54811527858936190292644189794, −2.03972722722047631783187160554, −1.52567497757072953058885138612,
1.52567497757072953058885138612, 2.03972722722047631783187160554, 3.54811527858936190292644189794, 4.32007388267191401416371115554, 5.23563368368958247950216116249, 6.19873567972955490648429131697, 6.76172135011007272509656671776, 7.32528635712878165456751149045, 8.789263205203508995070140004837, 9.108886333357813900499722783684
