| L(s) = 1 | + 2.71·3-s − 0.957·5-s − 1.45·7-s + 4.36·9-s + 4.83·11-s + 13-s − 2.59·15-s + 7.18·17-s + 4.48·19-s − 3.94·21-s + 1.00·23-s − 4.08·25-s + 3.71·27-s − 29-s − 2.69·31-s + 13.1·33-s + 1.39·35-s − 4.54·37-s + 2.71·39-s + 1.33·41-s − 2.40·43-s − 4.18·45-s + 2.47·47-s − 4.88·49-s + 19.5·51-s + 3.16·53-s − 4.62·55-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 0.428·5-s − 0.549·7-s + 1.45·9-s + 1.45·11-s + 0.277·13-s − 0.671·15-s + 1.74·17-s + 1.02·19-s − 0.861·21-s + 0.209·23-s − 0.816·25-s + 0.714·27-s − 0.185·29-s − 0.484·31-s + 2.28·33-s + 0.235·35-s − 0.747·37-s + 0.434·39-s + 0.208·41-s − 0.367·43-s − 0.623·45-s + 0.360·47-s − 0.697·49-s + 2.73·51-s + 0.434·53-s − 0.624·55-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
| \(L(1)\) |
\(\approx\) |
\(3.842222788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.842222788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 0.957T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 31 | \( 1 + 2.69T + 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 4.04T + 59T^{2} \) |
| 61 | \( 1 - 0.0266T + 61T^{2} \) |
| 67 | \( 1 - 4.16T + 67T^{2} \) |
| 71 | \( 1 - 4.29T + 71T^{2} \) |
| 73 | \( 1 - 0.151T + 73T^{2} \) |
| 79 | \( 1 - 0.682T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.042646148660984827884207073977, −7.54598573816630498728144860714, −6.91347867623451259467381131498, −6.04195030062199132213673582356, −5.15275087820482227979583838997, −3.90226132722459142192929476103, −3.60709237784670713949373868437, −3.06777767377506378898776795674, −1.90596348773427601961240664091, −1.02391410261818371240715619500,
1.02391410261818371240715619500, 1.90596348773427601961240664091, 3.06777767377506378898776795674, 3.60709237784670713949373868437, 3.90226132722459142192929476103, 5.15275087820482227979583838997, 6.04195030062199132213673582356, 6.91347867623451259467381131498, 7.54598573816630498728144860714, 8.042646148660984827884207073977
