| L(s) = 1 | + 1.64·3-s − 3.30·5-s + 3.78·7-s − 0.302·9-s + 5.42·11-s − 4.90·13-s − 5.42·15-s − 1.64·19-s + 6.21·21-s + 3.78·23-s + 5.90·25-s − 5.42·27-s − 3.69·29-s − 5.42·31-s + 8.90·33-s − 12.4·35-s − 4·37-s − 8.06·39-s + 9.21·41-s + 1.14·43-s + 1.00·45-s − 11.9·47-s + 7.30·49-s − 12.9·53-s − 17.9·55-s − 2.69·57-s − 15.1·59-s + ⋯ |
| L(s) = 1 | + 0.948·3-s − 1.47·5-s + 1.42·7-s − 0.100·9-s + 1.63·11-s − 1.36·13-s − 1.40·15-s − 0.376·19-s + 1.35·21-s + 0.788·23-s + 1.18·25-s − 1.04·27-s − 0.686·29-s − 0.974·31-s + 1.55·33-s − 2.11·35-s − 0.657·37-s − 1.29·39-s + 1.43·41-s + 0.174·43-s + 0.149·45-s − 1.74·47-s + 1.04·49-s − 1.77·53-s − 2.41·55-s − 0.357·57-s − 1.96·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 11 | \( 1 - 5.42T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 0.211T + 61T^{2} \) |
| 67 | \( 1 - 6.56T + 67T^{2} \) |
| 71 | \( 1 - 6.56T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.21T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.45000907203775447530139209543, −7.13481086751710502370967596820, −6.06873692220141279125288928178, −4.92578435757876080124996041693, −4.55143672505025554605310790368, −3.76107815986193213612582381865, −3.22858292739093965260504199586, −2.16822797539758946449981458077, −1.39669710122834918693589320928, 0,
1.39669710122834918693589320928, 2.16822797539758946449981458077, 3.22858292739093965260504199586, 3.76107815986193213612582381865, 4.55143672505025554605310790368, 4.92578435757876080124996041693, 6.06873692220141279125288928178, 7.13481086751710502370967596820, 7.45000907203775447530139209543
