L(s) = 1 | + 1.23·5-s − 2.35·7-s + 6.15·11-s + 13-s − 6.47·17-s − 5.25·19-s − 3.47·25-s − 4·29-s − 2.35·31-s − 2.90·35-s − 10.9·37-s − 5.23·41-s + 7.60·43-s − 9.06·47-s − 1.47·49-s + 4.94·53-s + 7.60·55-s + 9.06·59-s + 4.47·61-s + 1.23·65-s + 14.6·67-s + 6.15·71-s + 6·73-s − 14.4·77-s − 15.2·79-s − 3.24·83-s − 8.00·85-s + ⋯ |
L(s) = 1 | + 0.552·5-s − 0.888·7-s + 1.85·11-s + 0.277·13-s − 1.56·17-s − 1.20·19-s − 0.694·25-s − 0.742·29-s − 0.422·31-s − 0.491·35-s − 1.79·37-s − 0.817·41-s + 1.16·43-s − 1.32·47-s − 0.210·49-s + 0.679·53-s + 1.02·55-s + 1.17·59-s + 0.572·61-s + 0.153·65-s + 1.79·67-s + 0.730·71-s + 0.702·73-s − 1.64·77-s − 1.71·79-s − 0.356·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 6.15T + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 3.24T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.482725712054051609603139569077, −7.05045183938184102009637076715, −6.61919631666396024483654682308, −6.16724757284630787390874034066, −5.19941841125051133000863688364, −3.99518098342958234458478020447, −3.73296692716548125457878034643, −2.34139864730003675941115932200, −1.58644161714384221864988963730, 0,
1.58644161714384221864988963730, 2.34139864730003675941115932200, 3.73296692716548125457878034643, 3.99518098342958234458478020447, 5.19941841125051133000863688364, 6.16724757284630787390874034066, 6.61919631666396024483654682308, 7.05045183938184102009637076715, 8.482725712054051609603139569077