| L(s) = 1 | − 3-s + 2·5-s + 2·7-s − 2·9-s + 11-s − 4·13-s − 2·15-s + 2·17-s − 2·21-s − 6·23-s − 25-s + 5·27-s + 2·29-s − 10·31-s − 33-s + 4·35-s − 10·37-s + 4·39-s + 9·41-s − 4·43-s − 4·45-s − 6·47-s − 3·49-s − 2·51-s + 2·55-s − 15·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s − 0.516·15-s + 0.485·17-s − 0.436·21-s − 1.25·23-s − 1/5·25-s + 0.962·27-s + 0.371·29-s − 1.79·31-s − 0.174·33-s + 0.676·35-s − 1.64·37-s + 0.640·39-s + 1.40·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.269·55-s − 1.95·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9478506866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9478506866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.07697374323419, −13.41494699439376, −12.71197343414587, −12.29535716988593, −11.90897283247624, −11.38903439341625, −10.90814006002409, −10.39450849183009, −9.889190651233745, −9.449214826868936, −8.902566209847239, −8.322758328787423, −7.739359266065986, −7.315607707949236, −6.576775773930314, −6.055406630619428, −5.636593042184629, −5.097610985440767, −4.769923980065492, −3.918008558726898, −3.270550330918613, −2.496574412192927, −1.878497415788931, −1.462897566727882, −0.3045366340670019,
0.3045366340670019, 1.462897566727882, 1.878497415788931, 2.496574412192927, 3.270550330918613, 3.918008558726898, 4.769923980065492, 5.097610985440767, 5.636593042184629, 6.055406630619428, 6.576775773930314, 7.315607707949236, 7.739359266065986, 8.322758328787423, 8.902566209847239, 9.449214826868936, 9.889190651233745, 10.39450849183009, 10.90814006002409, 11.38903439341625, 11.90897283247624, 12.29535716988593, 12.71197343414587, 13.41494699439376, 14.07697374323419
