| L(s) = 1 | − 2.02·2-s + 2.62·3-s + 2.09·4-s − 5.30·6-s + 0.965·7-s − 0.192·8-s + 3.86·9-s + 5.49·12-s − 4.52·13-s − 1.95·14-s − 3.80·16-s − 3.33·17-s − 7.82·18-s − 3.27·19-s + 2.53·21-s + 3.36·23-s − 0.505·24-s + 9.15·26-s + 2.27·27-s + 2.02·28-s − 4.91·29-s − 0.418·31-s + 8.07·32-s + 6.75·34-s + 8.10·36-s − 6.33·37-s + 6.63·38-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.51·3-s + 1.04·4-s − 2.16·6-s + 0.365·7-s − 0.0681·8-s + 1.28·9-s + 1.58·12-s − 1.25·13-s − 0.522·14-s − 0.950·16-s − 0.809·17-s − 1.84·18-s − 0.751·19-s + 0.552·21-s + 0.701·23-s − 0.103·24-s + 1.79·26-s + 0.437·27-s + 0.382·28-s − 0.912·29-s − 0.0751·31-s + 1.42·32-s + 1.15·34-s + 1.35·36-s − 1.04·37-s + 1.07·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 - 0.965T + 7T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 + 0.418T + 31T^{2} \) |
| 37 | \( 1 + 6.33T + 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 2.66T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 - 9.60T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 1.43T + 73T^{2} \) |
| 79 | \( 1 + 1.00T + 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−8.472430988842624902672198698361, −7.891079943652713341710447122731, −7.22527250332356477552649733089, −6.66070272243461413580345763425, −5.10417837460218647838967079960, −4.31072693336891976137423150657, −3.19823428812774693873738068071, −2.22868990499284291557285656853, −1.69773286226106987729126884617, 0,
1.69773286226106987729126884617, 2.22868990499284291557285656853, 3.19823428812774693873738068071, 4.31072693336891976137423150657, 5.10417837460218647838967079960, 6.66070272243461413580345763425, 7.22527250332356477552649733089, 7.891079943652713341710447122731, 8.472430988842624902672198698361
