| L(s) = 1 | + 2·5-s − 4·7-s + 2·11-s + 6·17-s − 5·19-s − 8·23-s − 25-s − 31-s − 8·35-s + 7·37-s + 2·41-s + 43-s − 6·47-s + 9·49-s + 4·55-s + 6·59-s − 3·61-s − 3·67-s + 8·71-s + 11·73-s − 8·77-s − 3·79-s + 4·83-s + 12·85-s + 14·89-s − 10·95-s − 17·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s + 0.603·11-s + 1.45·17-s − 1.14·19-s − 1.66·23-s − 1/5·25-s − 0.179·31-s − 1.35·35-s + 1.15·37-s + 0.312·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s + 0.539·55-s + 0.781·59-s − 0.384·61-s − 0.366·67-s + 0.949·71-s + 1.28·73-s − 0.911·77-s − 0.337·79-s + 0.439·83-s + 1.30·85-s + 1.48·89-s − 1.02·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784869921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784869921\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.96810958598226, −13.68539882566567, −13.00187706637729, −12.68036226052401, −12.22725863909592, −11.72035572144959, −11.02837707649256, −10.32800100855322, −9.972897409831178, −9.623467740952206, −9.269377964526414, −8.528979921394976, −7.944707792818169, −7.422334715005093, −6.541219735013907, −6.329080789404513, −5.907407382834518, −5.391218476428426, −4.511245965540507, −3.762572850898646, −3.527772147356130, −2.609215225930966, −2.139419026773595, −1.339759157568113, −0.4418628626419893,
0.4418628626419893, 1.339759157568113, 2.139419026773595, 2.609215225930966, 3.527772147356130, 3.762572850898646, 4.511245965540507, 5.391218476428426, 5.907407382834518, 6.329080789404513, 6.541219735013907, 7.422334715005093, 7.944707792818169, 8.528979921394976, 9.269377964526414, 9.623467740952206, 9.972897409831178, 10.32800100855322, 11.02837707649256, 11.72035572144959, 12.22725863909592, 12.68036226052401, 13.00187706637729, 13.68539882566567, 13.96810958598226
