| L(s) = 1 | − 0.550·3-s − 5-s − 1.08·7-s − 2.69·9-s + 2.60·11-s + 1.03·13-s + 0.550·15-s − 4.29·17-s − 7.67·19-s + 0.596·21-s + 3.10·23-s + 25-s + 3.13·27-s − 4.16·29-s + 10.1·31-s − 1.43·33-s + 1.08·35-s + 37-s − 0.567·39-s − 0.331·41-s + 7.22·43-s + 2.69·45-s + 0.957·47-s − 5.82·49-s + 2.36·51-s − 8.95·53-s − 2.60·55-s + ⋯ |
| L(s) = 1 | − 0.317·3-s − 0.447·5-s − 0.409·7-s − 0.899·9-s + 0.784·11-s + 0.285·13-s + 0.142·15-s − 1.04·17-s − 1.76·19-s + 0.130·21-s + 0.646·23-s + 0.200·25-s + 0.603·27-s − 0.773·29-s + 1.82·31-s − 0.249·33-s + 0.183·35-s + 0.164·37-s − 0.0908·39-s − 0.0517·41-s + 1.10·43-s + 0.402·45-s + 0.139·47-s − 0.832·49-s + 0.331·51-s − 1.23·53-s − 0.350·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.017006274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017006274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 0.550T + 3T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 41 | \( 1 + 0.331T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 - 0.957T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 6.77T + 67T^{2} \) |
| 71 | \( 1 + 6.22T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 2.78T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 3.58T + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−8.663319535931010170780669311808, −8.228851025793524229852051379189, −7.03598161901266331986835278129, −6.43226678170785133331933527901, −5.90858728948990299479976708659, −4.72155712373662968952110595066, −4.10831661945571945661652163096, −3.11779423735335569896339903804, −2.14552724634703543335637460282, −0.60531509951207673363105434384,
0.60531509951207673363105434384, 2.14552724634703543335637460282, 3.11779423735335569896339903804, 4.10831661945571945661652163096, 4.72155712373662968952110595066, 5.90858728948990299479976708659, 6.43226678170785133331933527901, 7.03598161901266331986835278129, 8.228851025793524229852051379189, 8.663319535931010170780669311808
