| L(s) = 1 | − 0.329·5-s + 7-s − 1.32·11-s + 3.07·13-s − 7.35·17-s + 2.93·19-s − 6.68·23-s − 4.89·25-s + 7.76·29-s + 3.27·31-s − 0.329·35-s + 0.329·37-s − 0.271·41-s + 10.9·43-s + 1.14·47-s + 49-s − 6.42·53-s + 0.437·55-s + 0.744·59-s + 8.84·61-s − 1.01·65-s − 8.57·67-s + 1.60·71-s − 13.4·73-s − 1.32·77-s + 1.25·79-s − 0.0632·83-s + ⋯ |
| L(s) = 1 | − 0.147·5-s + 0.377·7-s − 0.400·11-s + 0.853·13-s − 1.78·17-s + 0.673·19-s − 1.39·23-s − 0.978·25-s + 1.44·29-s + 0.587·31-s − 0.0556·35-s + 0.0540·37-s − 0.0423·41-s + 1.67·43-s + 0.166·47-s + 0.142·49-s − 0.882·53-s + 0.0589·55-s + 0.0969·59-s + 1.13·61-s − 0.125·65-s − 1.04·67-s + 0.190·71-s − 1.57·73-s − 0.151·77-s + 0.141·79-s − 0.00694·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 5 | \( 1 + 0.329T + 5T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 + 0.271T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 - 1.14T + 47T^{2} \) |
| 53 | \( 1 + 6.42T + 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 - 8.84T + 61T^{2} \) |
| 67 | \( 1 + 8.57T + 67T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 1.25T + 79T^{2} \) |
| 83 | \( 1 + 0.0632T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.43279026237555912082000866330, −6.66578744891323923073997291563, −6.03850092993474033019467239989, −5.39847347473967546308232887883, −4.28803813610274511830501722288, −4.18689383892473664000184782848, −2.94565516525182439316079147245, −2.23248415019445111402167729080, −1.25690113769163489612446343164, 0,
1.25690113769163489612446343164, 2.23248415019445111402167729080, 2.94565516525182439316079147245, 4.18689383892473664000184782848, 4.28803813610274511830501722288, 5.39847347473967546308232887883, 6.03850092993474033019467239989, 6.66578744891323923073997291563, 7.43279026237555912082000866330
