| L(s) = 1 | + 18.6·2-s − 81·3-s − 165.·4-s + 1.49e3·5-s − 1.50e3·6-s + 1.18e4·7-s − 1.26e4·8-s + 6.56e3·9-s + 2.77e4·10-s + 5.41e4·11-s + 1.34e4·12-s + 1.76e5·13-s + 2.19e5·14-s − 1.20e5·15-s − 1.49e5·16-s + 1.08e5·17-s + 1.22e5·18-s + 5.44e4·19-s − 2.47e5·20-s − 9.55e5·21-s + 1.00e6·22-s − 2.05e6·23-s + 1.02e6·24-s + 2.75e5·25-s + 3.27e6·26-s − 5.31e5·27-s − 1.95e6·28-s + ⋯ |
| L(s) = 1 | + 0.822·2-s − 0.577·3-s − 0.323·4-s + 1.06·5-s − 0.474·6-s + 1.85·7-s − 1.08·8-s + 0.333·9-s + 0.878·10-s + 1.11·11-s + 0.186·12-s + 1.70·13-s + 1.52·14-s − 0.616·15-s − 0.571·16-s + 0.315·17-s + 0.274·18-s + 0.0958·19-s − 0.345·20-s − 1.07·21-s + 0.916·22-s − 1.52·23-s + 0.628·24-s + 0.140·25-s + 1.40·26-s − 0.192·27-s − 0.600·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.344578749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.344578749\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 - 18.6T + 512T^{2} \) |
| 5 | \( 1 - 1.49e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.18e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.41e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.76e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.08e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.44e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.05e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.15e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.67e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.48e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.73e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.96e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.91e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 9.09e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.83e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.08e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.21e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.20e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.48e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.18e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.57e9T + 7.60e17T^{2} \) |
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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
−11.38950425191940642608879118122, −10.12383527535197030702999300813, −8.985442195050720738117420299798, −8.059797125964045788956523195078, −6.23250783515934606918527121600, −5.75286280967338834327491120143, −4.66274349987443231142880820966, −3.80822395614208462352118407867, −1.86654002683893347293371294864, −1.05499059598594061148343710262,
1.05499059598594061148343710262, 1.86654002683893347293371294864, 3.80822395614208462352118407867, 4.66274349987443231142880820966, 5.75286280967338834327491120143, 6.23250783515934606918527121600, 8.059797125964045788956523195078, 8.985442195050720738117420299798, 10.12383527535197030702999300813, 11.38950425191940642608879118122
