| L(s) = 1 | + 5-s − 1.71·7-s − 2.52·11-s − 6.72·13-s − 4.44·17-s − 3.10·19-s − 23-s + 25-s − 5.01·29-s − 1.13·31-s − 1.71·35-s + 2.49·37-s + 7.39·41-s − 6.78·43-s + 13.3·47-s − 4.07·49-s + 12.4·53-s − 2.52·55-s − 11.0·59-s − 2.89·61-s − 6.72·65-s − 0.0836·67-s + 13.8·71-s − 5.57·73-s + 4.31·77-s − 1.97·83-s − 4.44·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.646·7-s − 0.761·11-s − 1.86·13-s − 1.07·17-s − 0.711·19-s − 0.208·23-s + 0.200·25-s − 0.932·29-s − 0.203·31-s − 0.289·35-s + 0.410·37-s + 1.15·41-s − 1.03·43-s + 1.94·47-s − 0.581·49-s + 1.70·53-s − 0.340·55-s − 1.43·59-s − 0.370·61-s − 0.834·65-s − 0.0102·67-s + 1.63·71-s − 0.652·73-s + 0.492·77-s − 0.216·83-s − 0.481·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9303375018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9303375018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 29 | \( 1 + 5.01T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 - 7.39T + 41T^{2} \) |
| 43 | \( 1 + 6.78T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 + 0.0836T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 - 9.83T + 97T^{2} \) |
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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
−7.50082474588683482384927277299, −7.31686399516980579205972332162, −6.36150455729099552207099402633, −5.81289421490829974512469391363, −4.95015872125196405763062557752, −4.43621020966856785960743333535, −3.41554949324568586926652411689, −2.37441077577317200600836487942, −2.17878685875724940933427396977, −0.43685126080614677780437826992,
0.43685126080614677780437826992, 2.17878685875724940933427396977, 2.37441077577317200600836487942, 3.41554949324568586926652411689, 4.43621020966856785960743333535, 4.95015872125196405763062557752, 5.81289421490829974512469391363, 6.36150455729099552207099402633, 7.31686399516980579205972332162, 7.50082474588683482384927277299
