L(s) = 1 | + 2.42·2-s − 1.86·3-s + 3.87·4-s + 2.06·5-s − 4.51·6-s − 4.60·7-s + 4.55·8-s + 0.464·9-s + 5.00·10-s − 2.98·11-s − 7.22·12-s + 0.123·13-s − 11.1·14-s − 3.83·15-s + 3.29·16-s − 1.25·17-s + 1.12·18-s − 1.44·19-s + 8.00·20-s + 8.57·21-s − 7.23·22-s − 5.31·23-s − 8.48·24-s − 0.747·25-s + 0.298·26-s + 4.71·27-s − 17.8·28-s + ⋯ |
L(s) = 1 | + 1.71·2-s − 1.07·3-s + 1.93·4-s + 0.922·5-s − 1.84·6-s − 1.74·7-s + 1.61·8-s + 0.154·9-s + 1.58·10-s − 0.900·11-s − 2.08·12-s + 0.0341·13-s − 2.98·14-s − 0.991·15-s + 0.823·16-s − 0.304·17-s + 0.265·18-s − 0.330·19-s + 1.78·20-s + 1.87·21-s − 1.54·22-s − 1.10·23-s − 1.73·24-s − 0.149·25-s + 0.0585·26-s + 0.908·27-s − 3.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 + 1.86T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.123T + 13T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 + 1.44T + 19T^{2} \) |
| 23 | \( 1 + 5.31T + 23T^{2} \) |
| 29 | \( 1 - 0.662T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 47 | \( 1 + 0.503T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + 8.70T + 59T^{2} \) |
| 61 | \( 1 + 2.85T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 0.0416T + 73T^{2} \) |
| 79 | \( 1 - 4.41T + 79T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−9.115852438356746721375837209083, −7.59176285346382730380052830103, −6.61979335926795269132161708190, −6.02204669947839427248275401771, −5.83007882410201572060681078909, −4.95741694429012570773866858671, −3.94892943321281431203180891166, −2.98963015485249900481974771114, −2.17819099609554718236809097250, 0,
2.17819099609554718236809097250, 2.98963015485249900481974771114, 3.94892943321281431203180891166, 4.95741694429012570773866858671, 5.83007882410201572060681078909, 6.02204669947839427248275401771, 6.61979335926795269132161708190, 7.59176285346382730380052830103, 9.115852438356746721375837209083