| L(s) = 1 | − 5-s − 7-s − 5·11-s − 5·13-s − 17-s + 5·19-s − 23-s − 4·25-s + 6·29-s + 6·31-s + 35-s + 4·37-s − 7·41-s + 7·43-s + 6·47-s + 49-s − 6·53-s + 5·55-s + 14·59-s + 5·65-s + 12·67-s + 4·71-s + 6·73-s + 5·77-s + 6·79-s − 6·83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.50·11-s − 1.38·13-s − 0.242·17-s + 1.14·19-s − 0.208·23-s − 4/5·25-s + 1.11·29-s + 1.07·31-s + 0.169·35-s + 0.657·37-s − 1.09·41-s + 1.06·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 0.674·55-s + 1.82·59-s + 0.620·65-s + 1.46·67-s + 0.474·71-s + 0.702·73-s + 0.569·77-s + 0.675·79-s − 0.658·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.82925436238397, −15.72541843067400, −15.29845241286284, −14.40647014447329, −13.96168340535684, −13.39542757886282, −12.76563622465282, −12.27147964060724, −11.77025325237115, −11.19522178404682, −10.40358981114215, −9.906327293307818, −9.634574412327307, −8.663908446685920, −7.980734332222263, −7.663420167626025, −7.020450579639074, −6.339202458687834, −5.425174476412974, −5.063825813706469, −4.337667807086211, −3.516188138723827, −2.626541804556756, −2.378339004749552, −0.8939672200017849, 0,
0.8939672200017849, 2.378339004749552, 2.626541804556756, 3.516188138723827, 4.337667807086211, 5.063825813706469, 5.425174476412974, 6.339202458687834, 7.020450579639074, 7.663420167626025, 7.980734332222263, 8.663908446685920, 9.634574412327307, 9.906327293307818, 10.40358981114215, 11.19522178404682, 11.77025325237115, 12.27147964060724, 12.76563622465282, 13.39542757886282, 13.96168340535684, 14.40647014447329, 15.29845241286284, 15.72541843067400, 15.82925436238397
