| L(s) = 1 | + 3.41·3-s + 2.69·5-s + 7-s + 8.65·9-s − 4.48·11-s − 4.84·13-s + 9.19·15-s + 2.43·19-s + 3.41·21-s + 0.501·23-s + 2.26·25-s + 19.2·27-s + 0.725·29-s + 9.53·31-s − 15.2·33-s + 2.69·35-s + 6.71·37-s − 16.5·39-s − 3.01·41-s + 7.93·43-s + 23.3·45-s + 2.74·47-s + 49-s + 1.98·53-s − 12.0·55-s + 8.32·57-s − 0.927·59-s + ⋯ |
| L(s) = 1 | + 1.97·3-s + 1.20·5-s + 0.377·7-s + 2.88·9-s − 1.35·11-s − 1.34·13-s + 2.37·15-s + 0.559·19-s + 0.744·21-s + 0.104·23-s + 0.452·25-s + 3.71·27-s + 0.134·29-s + 1.71·31-s − 2.66·33-s + 0.455·35-s + 1.10·37-s − 2.65·39-s − 0.471·41-s + 1.21·43-s + 3.47·45-s + 0.399·47-s + 0.142·49-s + 0.273·53-s − 1.62·55-s + 1.10·57-s − 0.120·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.852730034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.852730034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 3.41T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 + 4.48T + 11T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 - 0.501T + 23T^{2} \) |
| 29 | \( 1 - 0.725T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 6.71T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 - 7.93T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 - 1.98T + 53T^{2} \) |
| 59 | \( 1 + 0.927T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 2.91T + 71T^{2} \) |
| 73 | \( 1 - 0.267T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.73391856880202214280109022889, −7.54136617314487741332690532248, −6.65506821397203659716146847476, −5.65509008390847226566420178286, −4.85150672329417503135389867539, −4.32337364292869710960555805539, −3.10130959853354998044658836994, −2.52715868093027594227449553502, −2.22454072501917530601237586183, −1.15075900641346740619424469823,
1.15075900641346740619424469823, 2.22454072501917530601237586183, 2.52715868093027594227449553502, 3.10130959853354998044658836994, 4.32337364292869710960555805539, 4.85150672329417503135389867539, 5.65509008390847226566420178286, 6.65506821397203659716146847476, 7.54136617314487741332690532248, 7.73391856880202214280109022889
