| L(s) = 1 | + 1.35·2-s + 2.22·3-s − 0.160·4-s − 2.22·5-s + 3.02·6-s − 2.92·8-s + 1.96·9-s − 3.02·10-s − 1.12·11-s − 0.356·12-s − 2.22·13-s − 4.96·15-s − 3.65·16-s − 4.05·17-s + 2.66·18-s + 3.61·19-s + 0.356·20-s − 1.53·22-s − 2.68·23-s − 6.52·24-s − 0.0364·25-s − 3.02·26-s − 2.30·27-s − 3.72·29-s − 6.73·30-s − 0.345·31-s + 0.902·32-s + ⋯ |
| L(s) = 1 | + 0.959·2-s + 1.28·3-s − 0.0800·4-s − 0.996·5-s + 1.23·6-s − 1.03·8-s + 0.654·9-s − 0.955·10-s − 0.340·11-s − 0.102·12-s − 0.617·13-s − 1.28·15-s − 0.913·16-s − 0.983·17-s + 0.627·18-s + 0.829·19-s + 0.0797·20-s − 0.326·22-s − 0.560·23-s − 1.33·24-s − 0.00729·25-s − 0.592·26-s − 0.444·27-s − 0.691·29-s − 1.22·30-s − 0.0620·31-s + 0.159·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.35T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 2.22T + 13T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 + 3.72T + 29T^{2} \) |
| 31 | \( 1 + 0.345T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 5.22T + 59T^{2} \) |
| 61 | \( 1 + 5.60T + 61T^{2} \) |
| 67 | \( 1 + 8.83T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 0.815T + 73T^{2} \) |
| 79 | \( 1 + 8.84T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.746068558065838339195888579542, −7.919502354345474807220466848177, −7.44872902538964584837048134692, −6.34384438943154810935136201842, −5.26451577702681737435381246733, −4.45339515685719469401413940419, −3.72893893017871731861718286643, −3.09171513384321421615814405059, −2.17459746040910271279346551381, 0,
2.17459746040910271279346551381, 3.09171513384321421615814405059, 3.72893893017871731861718286643, 4.45339515685719469401413940419, 5.26451577702681737435381246733, 6.34384438943154810935136201842, 7.44872902538964584837048134692, 7.919502354345474807220466848177, 8.746068558065838339195888579542
