| L(s) = 1 | − 2.02·3-s − 3.01·5-s + 3.00·7-s + 1.10·9-s − 3.74·11-s − 13-s + 6.10·15-s + 1.06·17-s − 1.48·19-s − 6.08·21-s + 4.25·23-s + 4.09·25-s + 3.84·27-s − 29-s − 7.29·31-s + 7.58·33-s − 9.05·35-s + 7.39·37-s + 2.02·39-s + 10.0·41-s − 4.85·43-s − 3.32·45-s − 3.52·47-s + 2.01·49-s − 2.15·51-s + 10.0·53-s + 11.2·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s − 1.34·5-s + 1.13·7-s + 0.367·9-s − 1.12·11-s − 0.277·13-s + 1.57·15-s + 0.258·17-s − 0.339·19-s − 1.32·21-s + 0.887·23-s + 0.818·25-s + 0.739·27-s − 0.185·29-s − 1.30·31-s + 1.32·33-s − 1.53·35-s + 1.21·37-s + 0.324·39-s + 1.57·41-s − 0.740·43-s − 0.495·45-s − 0.513·47-s + 0.287·49-s − 0.302·51-s + 1.38·53-s + 1.52·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 - 3.00T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 + 3.52T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.35T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 4.35T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 9.18T + 79T^{2} \) |
| 83 | \( 1 - 0.299T + 83T^{2} \) |
| 89 | \( 1 - 5.16T + 89T^{2} \) |
| 97 | \( 1 + 7.20T + 97T^{2} \) |
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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
−7.56760456664158657018228096124, −7.29135524153332456619632247415, −6.18139268568466535251158324230, −5.42532387449599109959689416779, −4.87101831732623835902492838713, −4.33661003464051245249491552695, −3.33695588308034256247848897759, −2.29285377416583059095219107776, −0.950031592526461356321205677635, 0,
0.950031592526461356321205677635, 2.29285377416583059095219107776, 3.33695588308034256247848897759, 4.33661003464051245249491552695, 4.87101831732623835902492838713, 5.42532387449599109959689416779, 6.18139268568466535251158324230, 7.29135524153332456619632247415, 7.56760456664158657018228096124
