| L(s) = 1 | + 1.63·2-s + 0.665·4-s − 1.75·5-s + 1.29·7-s − 2.17·8-s − 2.86·10-s − 2.05·11-s − 1.58·13-s + 2.10·14-s − 4.88·16-s − 4.57·17-s + 5.81·19-s − 1.16·20-s − 3.34·22-s − 0.643·23-s − 1.91·25-s − 2.58·26-s + 0.858·28-s − 7.32·29-s − 1.52·31-s − 3.62·32-s − 7.47·34-s − 2.26·35-s − 8.16·37-s + 9.48·38-s + 3.82·40-s − 6.76·41-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.332·4-s − 0.785·5-s + 0.487·7-s − 0.770·8-s − 0.907·10-s − 0.618·11-s − 0.438·13-s + 0.563·14-s − 1.22·16-s − 1.11·17-s + 1.33·19-s − 0.261·20-s − 0.713·22-s − 0.134·23-s − 0.382·25-s − 0.506·26-s + 0.162·28-s − 1.36·29-s − 0.273·31-s − 0.640·32-s − 1.28·34-s − 0.383·35-s − 1.34·37-s + 1.53·38-s + 0.605·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 - 1.29T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 - 5.81T + 19T^{2} \) |
| 23 | \( 1 + 0.643T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 - 0.500T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 0.214T + 61T^{2} \) |
| 67 | \( 1 - 4.60T + 67T^{2} \) |
| 71 | \( 1 - 5.57T + 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 - 4.03T + 79T^{2} \) |
| 83 | \( 1 + 9.57T + 83T^{2} \) |
| 89 | \( 1 + 4.61T + 89T^{2} \) |
| 97 | \( 1 - 8.69T + 97T^{2} \) |
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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
−9.154694601229226752834435914566, −8.297423697098702169262057807971, −7.47543755114678560432548262223, −6.69210480497328062573485997277, −5.42848478788440722278889806967, −5.04055575038137767117223996175, −4.02740479605548020350961062118, −3.33844372582244043153359645790, −2.12647758027134238541723185355, 0,
2.12647758027134238541723185355, 3.33844372582244043153359645790, 4.02740479605548020350961062118, 5.04055575038137767117223996175, 5.42848478788440722278889806967, 6.69210480497328062573485997277, 7.47543755114678560432548262223, 8.297423697098702169262057807971, 9.154694601229226752834435914566
