| L(s) = 1 | − 0.445·5-s + 9-s − 1.24·11-s + 1.24·17-s + 1.80·19-s − 0.801·25-s − 1.80·29-s + 0.445·31-s + 1.24·37-s + 0.445·43-s − 0.445·45-s + 1.80·47-s + 49-s + 0.554·55-s + 0.445·59-s + 81-s − 0.554·85-s − 0.801·95-s − 1.80·97-s − 1.24·99-s − 1.24·103-s + ⋯ |
| L(s) = 1 | − 0.445·5-s + 9-s − 1.24·11-s + 1.24·17-s + 1.80·19-s − 0.801·25-s − 1.80·29-s + 0.445·31-s + 1.24·37-s + 0.445·43-s − 0.445·45-s + 1.80·47-s + 49-s + 0.554·55-s + 0.445·59-s + 81-s − 0.554·85-s − 0.801·95-s − 1.80·97-s − 1.24·99-s − 1.24·103-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.178593228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178593228\) |
| \(L(1)\) |
|
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 | \( 1 + 0.445T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.24T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 - 1.80T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.80T + T^{2} \) |
| 31 | \( 1 - 0.445T + T^{2} \) |
| 37 | \( 1 - 1.24T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.445T + T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.445T + T^{2} \) |
| 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 | \( 1 + 1.80T + T^{2} \) |
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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
−9.369702763780397125463022943937, −8.092417378572260534345722013978, −7.53287059693151949237330053042, −7.24479411635748601297436027117, −5.76380618875045537725955331944, −5.36800903502916087727615444234, −4.25411165663842580277392405999, −3.49370919076876755567305713779, −2.46259916971146280555814577389, −1.09232386149137611520252174720,
1.09232386149137611520252174720, 2.46259916971146280555814577389, 3.49370919076876755567305713779, 4.25411165663842580277392405999, 5.36800903502916087727615444234, 5.76380618875045537725955331944, 7.24479411635748601297436027117, 7.53287059693151949237330053042, 8.092417378572260534345722013978, 9.369702763780397125463022943937
