| L(s) = 1 | + 3-s − 0.853·5-s − 3.83·7-s + 9-s + 3.34·11-s − 3.14·13-s − 0.853·15-s + 3.63·17-s − 1.14·19-s − 3.83·21-s − 2.85·23-s − 4.27·25-s + 27-s + 8.02·29-s + 9.86·31-s + 3.34·33-s + 3.27·35-s − 8.19·37-s − 3.14·39-s + 41-s − 11.7·43-s − 0.853·45-s + 8.32·47-s + 7.68·49-s + 3.63·51-s − 1.60·53-s − 2.85·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.381·5-s − 1.44·7-s + 0.333·9-s + 1.00·11-s − 0.872·13-s − 0.220·15-s + 0.881·17-s − 0.262·19-s − 0.836·21-s − 0.595·23-s − 0.854·25-s + 0.192·27-s + 1.49·29-s + 1.77·31-s + 0.581·33-s + 0.552·35-s − 1.34·37-s − 0.503·39-s + 0.156·41-s − 1.79·43-s − 0.127·45-s + 1.21·47-s + 1.09·49-s + 0.509·51-s − 0.220·53-s − 0.384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 5 | \( 1 + 0.853T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 - 3.63T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 - 9.86T + 31T^{2} \) |
| 37 | \( 1 + 8.19T + 37T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−7.47438804608153778197210412910, −6.74026172494433305958548217273, −6.38853025900269941530317261545, −5.45253994608673946508596959092, −4.44144405175992620302792869265, −3.81257109696915791504804472533, −3.13768792559797386318107109613, −2.46572499069880589673703413298, −1.23848205244003722031414181916, 0,
1.23848205244003722031414181916, 2.46572499069880589673703413298, 3.13768792559797386318107109613, 3.81257109696915791504804472533, 4.44144405175992620302792869265, 5.45253994608673946508596959092, 6.38853025900269941530317261545, 6.74026172494433305958548217273, 7.47438804608153778197210412910
