| L(s) = 1 | − 7·7-s + 9·9-s + 6·11-s + 18·23-s + 25·25-s + 54·29-s + 38·37-s − 58·43-s + 49·49-s + 6·53-s − 63·63-s + 118·67-s + 114·71-s − 42·77-s − 94·79-s + 81·81-s + 54·99-s − 186·107-s − 106·109-s − 222·113-s + ⋯ |
| L(s) = 1 | − 7-s + 9-s + 6/11·11-s + 0.782·23-s + 25-s + 1.86·29-s + 1.02·37-s − 1.34·43-s + 49-s + 6/53·53-s − 63-s + 1.76·67-s + 1.60·71-s − 0.545·77-s − 1.18·79-s + 81-s + 6/11·99-s − 1.73·107-s − 0.972·109-s − 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.740784986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740784986\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( 1 - 6 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 18 T + p^{2} T^{2} \) |
| 29 | \( 1 - 54 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 38 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 58 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 6 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( 1 - 118 T + p^{2} T^{2} \) |
| 71 | \( 1 - 114 T + p^{2} T^{2} \) |
| 73 | \( ( 1 - p T )( 1 + p T ) \) |
| 79 | \( 1 + 94 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( ( 1 - p T )( 1 + p T ) \) |
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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
−10.73830918317970545877288914776, −9.941189933791150644797200698557, −9.243419839209355245147620139421, −8.199905250505154561781615266109, −6.88132869855223367837814385374, −6.52801920420788727038954527995, −5.04620871040743103669404843060, −3.96498699796848182717686340118, −2.79938302349374874815625336341, −1.04275533797669960242719035770,
1.04275533797669960242719035770, 2.79938302349374874815625336341, 3.96498699796848182717686340118, 5.04620871040743103669404843060, 6.52801920420788727038954527995, 6.88132869855223367837814385374, 8.199905250505154561781615266109, 9.243419839209355245147620139421, 9.941189933791150644797200698557, 10.73830918317970545877288914776
