| L(s) = 1 | + 2.45·2-s + 4.03·4-s + 5-s − 3.28·7-s + 4.98·8-s + 2.45·10-s + 0.313·13-s − 8.07·14-s + 4.18·16-s + 5·17-s + 7.45·19-s + 4.03·20-s − 1.07·23-s + 25-s + 0.769·26-s − 13.2·28-s − 5.03·29-s + 3.44·31-s + 0.310·32-s + 12.2·34-s − 3.28·35-s + 2.63·37-s + 18.3·38-s + 4.98·40-s + 10.8·41-s + 5.51·43-s − 2.63·46-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.01·4-s + 0.447·5-s − 1.24·7-s + 1.76·8-s + 0.776·10-s + 0.0868·13-s − 2.15·14-s + 1.04·16-s + 1.21·17-s + 1.71·19-s + 0.901·20-s − 0.223·23-s + 0.200·25-s + 0.150·26-s − 2.50·28-s − 0.935·29-s + 0.619·31-s + 0.0549·32-s + 2.10·34-s − 0.555·35-s + 0.433·37-s + 2.96·38-s + 0.788·40-s + 1.69·41-s + 0.840·43-s − 0.388·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.107559134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.107559134\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 13 | \( 1 - 0.313T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 2.63T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 + 9.18T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 - 8.65T + 73T^{2} \) |
| 79 | \( 1 + 5.41T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 - 0.224T + 97T^{2} \) |
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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
−7.61375027402244834396143620643, −7.35793383097908001769903632368, −6.32779564481636949984260987095, −5.85410497763592177975276079214, −5.43369856039735912384871515937, −4.46463152913180617158482958190, −3.62883832362202195624116019429, −3.09310119581155121605901356563, −2.40844186547185241948732526093, −1.06119113990872566687886488534,
1.06119113990872566687886488534, 2.40844186547185241948732526093, 3.09310119581155121605901356563, 3.62883832362202195624116019429, 4.46463152913180617158482958190, 5.43369856039735912384871515937, 5.85410497763592177975276079214, 6.32779564481636949984260987095, 7.35793383097908001769903632368, 7.61375027402244834396143620643
