| L(s) = 1 | − 2.22·3-s − 3.90·5-s − 0.127·7-s + 1.95·9-s + 1.34·11-s − 1.48·13-s + 8.70·15-s + 1.12·17-s − 3.57·19-s + 0.283·21-s − 6.17·23-s + 10.2·25-s + 2.31·27-s − 0.0940·29-s + 1.35·31-s − 3.00·33-s + 0.497·35-s + 2.38·37-s + 3.31·39-s + 9.91·41-s + 2.37·43-s − 7.65·45-s + 12.8·47-s − 6.98·49-s − 2.50·51-s + 0.100·53-s − 5.27·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 1.74·5-s − 0.0480·7-s + 0.652·9-s + 0.406·11-s − 0.412·13-s + 2.24·15-s + 0.273·17-s − 0.819·19-s + 0.0618·21-s − 1.28·23-s + 2.05·25-s + 0.446·27-s − 0.0174·29-s + 0.243·31-s − 0.522·33-s + 0.0840·35-s + 0.392·37-s + 0.530·39-s + 1.54·41-s + 0.362·43-s − 1.14·45-s + 1.87·47-s − 0.997·49-s − 0.351·51-s + 0.0137·53-s − 0.710·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.22T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.127T + 7T^{2} \) |
| 11 | \( 1 - 1.34T + 11T^{2} \) |
| 13 | \( 1 + 1.48T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 0.0940T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 9.91T + 41T^{2} \) |
| 43 | \( 1 - 2.37T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 0.100T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 - 5.58T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 - 4.22T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + 0.492T + 83T^{2} \) |
| 89 | \( 1 - 8.78T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.900119670438032490247909943205, −7.39320604823964802246202163929, −6.53104797957767566582875443257, −5.94933381695991699254504462394, −4.99207976752931560398734498939, −4.25405714713192018061884827748, −3.77875659999198636500732657882, −2.52648854920735691081075744839, −0.901230062955608843838096251287, 0,
0.901230062955608843838096251287, 2.52648854920735691081075744839, 3.77875659999198636500732657882, 4.25405714713192018061884827748, 4.99207976752931560398734498939, 5.94933381695991699254504462394, 6.53104797957767566582875443257, 7.39320604823964802246202163929, 7.900119670438032490247909943205
