| L(s) = 1 | − 5.48·3-s − 188.·7-s − 212.·9-s + 501.·11-s − 1.06e3·13-s + 29.5·17-s − 1.57e3·19-s + 1.03e3·21-s − 1.29e3·23-s + 2.50e3·27-s − 3.58e3·29-s + 3.52e3·31-s − 2.75e3·33-s − 8.41e3·37-s + 5.82e3·39-s + 7.01e3·41-s + 2.26e4·43-s − 3.50e3·47-s + 1.89e4·49-s − 162.·51-s − 2.73e4·53-s + 8.66e3·57-s − 7.92e3·59-s − 7.02e3·61-s + 4.02e4·63-s + 1.76e4·67-s + 7.11e3·69-s + ⋯ |
| L(s) = 1 | − 0.352·3-s − 1.45·7-s − 0.875·9-s + 1.25·11-s − 1.74·13-s + 0.0248·17-s − 1.00·19-s + 0.513·21-s − 0.510·23-s + 0.660·27-s − 0.791·29-s + 0.659·31-s − 0.440·33-s − 1.01·37-s + 0.613·39-s + 0.651·41-s + 1.87·43-s − 0.231·47-s + 1.12·49-s − 0.00874·51-s − 1.33·53-s + 0.353·57-s − 0.296·59-s − 0.241·61-s + 1.27·63-s + 0.479·67-s + 0.179·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6635209891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6635209891\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.48T + 243T^{2} \) |
| 7 | \( 1 + 188.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.06e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 29.5T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.57e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.50e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.84e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5T + 8.58e9T^{2} \) |
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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
−10.36244987390296642919466458675, −9.521692910259689599280000553426, −8.889854252283819897495031788363, −7.50123233469231067560634488191, −6.51867942475155803794290788988, −5.90848769032715316426031084139, −4.56825412070667402102041410507, −3.38592859200612421262858333424, −2.28272551085774623863229001013, −0.41158521510147163943682755608,
0.41158521510147163943682755608, 2.28272551085774623863229001013, 3.38592859200612421262858333424, 4.56825412070667402102041410507, 5.90848769032715316426031084139, 6.51867942475155803794290788988, 7.50123233469231067560634488191, 8.889854252283819897495031788363, 9.521692910259689599280000553426, 10.36244987390296642919466458675
