| L(s) = 1 | − 1.80·5-s + 9-s + 0.445·11-s − 0.445·17-s − 1.24·19-s + 2.24·25-s + 1.24·29-s + 1.80·31-s − 0.445·37-s + 1.80·43-s − 1.80·45-s − 1.24·47-s + 49-s − 0.801·55-s + 1.80·59-s + 81-s + 0.801·85-s + 2.24·95-s + 1.24·97-s + 0.445·99-s + 0.445·103-s + ⋯ |
| L(s) = 1 | − 1.80·5-s + 9-s + 0.445·11-s − 0.445·17-s − 1.24·19-s + 2.24·25-s + 1.24·29-s + 1.80·31-s − 0.445·37-s + 1.80·43-s − 1.80·45-s − 1.24·47-s + 49-s − 0.801·55-s + 1.80·59-s + 81-s + 0.801·85-s + 2.24·95-s + 1.24·97-s + 0.445·99-s + 0.445·103-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9138312221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9138312221\) |
| \(L(1)\) |
|
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 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.445T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.80T + T^{2} \) |
| 47 | \( 1 + 1.24T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 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 | \( 1 - 1.24T + T^{2} \) |
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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
−8.860142775791843156314434481298, −8.366841302616287259399977518022, −7.63668579183497951878923889820, −6.89204507256753160557068176208, −6.34608731573479789032513314538, −4.79176879696416914055573015322, −4.30253299427879068965112830313, −3.69105027374563071393202045430, −2.50698532742251492372225532607, −0.928052866426893134621527046383,
0.928052866426893134621527046383, 2.50698532742251492372225532607, 3.69105027374563071393202045430, 4.30253299427879068965112830313, 4.79176879696416914055573015322, 6.34608731573479789032513314538, 6.89204507256753160557068176208, 7.63668579183497951878923889820, 8.366841302616287259399977518022, 8.860142775791843156314434481298
