L(s) = 1 | − 2.52·2-s + 4.37·4-s − 5-s − 0.151·7-s − 5.99·8-s + 2.52·10-s + 2.63·11-s + 6.62·13-s + 0.383·14-s + 6.38·16-s − 3.24·17-s + 0.214·19-s − 4.37·20-s − 6.65·22-s + 4.81·23-s + 25-s − 16.7·26-s − 0.664·28-s − 9.81·29-s − 8.03·31-s − 4.12·32-s + 8.20·34-s + 0.151·35-s − 7.35·37-s − 0.542·38-s + 5.99·40-s − 11.8·41-s + ⋯ |
L(s) = 1 | − 1.78·2-s + 2.18·4-s − 0.447·5-s − 0.0574·7-s − 2.11·8-s + 0.798·10-s + 0.795·11-s + 1.83·13-s + 0.102·14-s + 1.59·16-s − 0.788·17-s + 0.0492·19-s − 0.978·20-s − 1.41·22-s + 1.00·23-s + 0.200·25-s − 3.28·26-s − 0.125·28-s − 1.82·29-s − 1.44·31-s − 0.730·32-s + 1.40·34-s + 0.0256·35-s − 1.20·37-s − 0.0879·38-s + 0.947·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 7 | \( 1 + 0.151T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 - 6.62T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 0.214T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 + 9.81T + 29T^{2} \) |
| 31 | \( 1 + 8.03T + 31T^{2} \) |
| 37 | \( 1 + 7.35T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 6.83T + 43T^{2} \) |
| 47 | \( 1 - 0.501T + 47T^{2} \) |
| 53 | \( 1 + 5.08T + 53T^{2} \) |
| 59 | \( 1 - 9.75T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 9.27T + 67T^{2} \) |
| 71 | \( 1 - 2.89T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−8.361587508769292481717033031386, −7.45197463607218659178967777865, −6.88418076388638785606993839775, −6.29565836249964556615416575663, −5.33100210307579823230648667086, −3.91409145047595698373516102344, −3.33131227904953169785127670159, −1.93295989544444428419562358463, −1.25750420333711526850505980132, 0,
1.25750420333711526850505980132, 1.93295989544444428419562358463, 3.33131227904953169785127670159, 3.91409145047595698373516102344, 5.33100210307579823230648667086, 6.29565836249964556615416575663, 6.88418076388638785606993839775, 7.45197463607218659178967777865, 8.361587508769292481717033031386