L(s) = 1 | − 3-s + 5-s + 9-s − 11-s − 5·13-s − 15-s + 2·17-s − 19-s + 2·23-s + 25-s − 27-s + 6·29-s − 31-s + 33-s + 37-s + 5·39-s + 8·41-s + 43-s + 45-s + 2·47-s − 2·51-s − 4·53-s − 55-s + 57-s + 6·59-s − 14·61-s − 5·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.179·31-s + 0.174·33-s + 0.164·37-s + 0.800·39-s + 1.24·41-s + 0.152·43-s + 0.149·45-s + 0.291·47-s − 0.280·51-s − 0.549·53-s − 0.134·55-s + 0.132·57-s + 0.781·59-s − 1.79·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.23324034895885, −14.80767608734396, −14.25319686540705, −13.77532313095351, −13.10813073520318, −12.48343299989571, −12.33238972817372, −11.59996925178836, −11.06496657094033, −10.29419241450827, −10.20760765797594, −9.402346974365497, −9.025642535678872, −8.157007172995611, −7.582324449634893, −7.100620612442994, −6.470658169381880, −5.786517326133449, −5.385244199672510, −4.592429910611511, −4.326454675159875, −3.100923749435616, −2.672810180952757, −1.819259102305770, −0.9590146692747762, 0,
0.9590146692747762, 1.819259102305770, 2.672810180952757, 3.100923749435616, 4.326454675159875, 4.592429910611511, 5.385244199672510, 5.786517326133449, 6.470658169381880, 7.100620612442994, 7.582324449634893, 8.157007172995611, 9.025642535678872, 9.402346974365497, 10.20760765797594, 10.29419241450827, 11.06496657094033, 11.59996925178836, 12.33238972817372, 12.48343299989571, 13.10813073520318, 13.77532313095351, 14.25319686540705, 14.80767608734396, 15.23324034895885