| L(s) = 1 | − 9·3-s + 160·7-s + 81·9-s − 596·11-s + 122·13-s + 1.07e3·17-s + 796·19-s − 1.44e3·21-s + 1.08e3·23-s − 729·27-s + 46·29-s − 4.95e3·31-s + 5.36e3·33-s + 6.11e3·37-s − 1.09e3·39-s − 6·41-s + 2.41e4·43-s − 1.34e4·47-s + 8.79e3·49-s − 9.70e3·51-s − 2.05e4·53-s − 7.16e3·57-s − 4.67e4·59-s − 9.60e3·61-s + 1.29e4·63-s + 1.74e4·67-s − 9.79e3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.23·7-s + 1/3·9-s − 1.48·11-s + 0.200·13-s + 0.904·17-s + 0.505·19-s − 0.712·21-s + 0.428·23-s − 0.192·27-s + 0.0101·29-s − 0.925·31-s + 0.857·33-s + 0.734·37-s − 0.115·39-s − 0.000557·41-s + 1.98·43-s − 0.890·47-s + 0.523·49-s − 0.522·51-s − 1.00·53-s − 0.292·57-s − 1.74·59-s − 0.330·61-s + 0.411·63-s + 0.473·67-s − 0.247·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.889757100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.889757100\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 160 T + p^{5} T^{2} \) |
| 11 | \( 1 + 596 T + p^{5} T^{2} \) |
| 13 | \( 1 - 122 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1078 T + p^{5} T^{2} \) |
| 19 | \( 1 - 796 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1088 T + p^{5} T^{2} \) |
| 29 | \( 1 - 46 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4952 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6114 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6 T + p^{5} T^{2} \) |
| 43 | \( 1 - 24116 T + p^{5} T^{2} \) |
| 47 | \( 1 + 13480 T + p^{5} T^{2} \) |
| 53 | \( 1 + 20598 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46756 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 17404 T + p^{5} T^{2} \) |
| 71 | \( 1 - 26568 T + p^{5} T^{2} \) |
| 73 | \( 1 + 75450 T + p^{5} T^{2} \) |
| 79 | \( 1 - 50472 T + p^{5} T^{2} \) |
| 83 | \( 1 + 33236 T + p^{5} T^{2} \) |
| 89 | \( 1 - 133194 T + p^{5} T^{2} \) |
| 97 | \( 1 - 42878 T + p^{5} T^{2} \) |
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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
−10.08440328236203547300207458633, −9.009726542165932015827347259018, −7.75200736924549397896838519065, −7.61263854758241228592848050033, −6.04627138683383733585836325597, −5.25204493797301607811032300958, −4.59051548108154405697701411926, −3.13908949074049935725557939115, −1.84005812748791400887809234109, −0.70145077181122262909665879081,
0.70145077181122262909665879081, 1.84005812748791400887809234109, 3.13908949074049935725557939115, 4.59051548108154405697701411926, 5.25204493797301607811032300958, 6.04627138683383733585836325597, 7.61263854758241228592848050033, 7.75200736924549397896838519065, 9.009726542165932015827347259018, 10.08440328236203547300207458633
