L(s) = 1 | + 1.80·2-s + 1.26·4-s + 2.14·5-s − 7-s − 1.32·8-s + 3.88·10-s − 3.66·13-s − 1.80·14-s − 4.92·16-s + 1.89·17-s + 4.44·19-s + 2.71·20-s − 5.20·23-s − 0.381·25-s − 6.62·26-s − 1.26·28-s − 3.76·29-s + 2.21·31-s − 6.25·32-s + 3.41·34-s − 2.14·35-s − 8.31·37-s + 8.03·38-s − 2.85·40-s + 0.0621·41-s − 8.99·43-s − 9.41·46-s + ⋯ |
L(s) = 1 | + 1.27·2-s + 0.632·4-s + 0.961·5-s − 0.377·7-s − 0.469·8-s + 1.22·10-s − 1.01·13-s − 0.482·14-s − 1.23·16-s + 0.458·17-s + 1.02·19-s + 0.607·20-s − 1.08·23-s − 0.0763·25-s − 1.29·26-s − 0.239·28-s − 0.698·29-s + 0.397·31-s − 1.10·32-s + 0.586·34-s − 0.363·35-s − 1.36·37-s + 1.30·38-s − 0.451·40-s + 0.00970·41-s − 1.37·43-s − 1.38·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.80T + 2T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 0.0621T + 41T^{2} \) |
| 43 | \( 1 + 8.99T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 5.13T + 61T^{2} \) |
| 67 | \( 1 + 9.38T + 67T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 - 7.65T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 + 6.68T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.28463492548130460262511602445, −6.59876391273794891193141625948, −5.85848675896668807961973844968, −5.44380460894904712417445030613, −4.83065035203394819788366657846, −3.94310497110004895005470973741, −3.22096629137690823096974429351, −2.49538928921549486083999091543, −1.64361692053921271140694395098, 0,
1.64361692053921271140694395098, 2.49538928921549486083999091543, 3.22096629137690823096974429351, 3.94310497110004895005470973741, 4.83065035203394819788366657846, 5.44380460894904712417445030613, 5.85848675896668807961973844968, 6.59876391273794891193141625948, 7.28463492548130460262511602445