| L(s) = 1 | + 2.23e5·5-s + 3.56e6·7-s + 8.12e7·11-s + 3.44e8·13-s − 2.43e9·17-s + 3.23e9·19-s − 1.26e10·23-s + 1.92e10·25-s − 9.41e10·29-s + 7.80e10·31-s + 7.95e11·35-s − 4.06e11·37-s + 2.98e11·41-s − 1.91e12·43-s − 5.51e11·47-s + 7.97e12·49-s + 1.20e13·53-s + 1.81e13·55-s − 6.58e12·59-s − 4.21e12·61-s + 7.68e13·65-s − 5.45e13·67-s + 1.17e14·71-s + 1.31e14·73-s + 2.89e14·77-s + 2.57e14·79-s − 3.45e14·83-s + ⋯ |
| L(s) = 1 | + 1.27·5-s + 1.63·7-s + 1.25·11-s + 1.52·13-s − 1.44·17-s + 0.829·19-s − 0.774·23-s + 0.630·25-s − 1.01·29-s + 0.509·31-s + 2.08·35-s − 0.703·37-s + 0.239·41-s − 1.07·43-s − 0.158·47-s + 1.67·49-s + 1.41·53-s + 1.60·55-s − 0.344·59-s − 0.171·61-s + 1.94·65-s − 1.09·67-s + 1.53·71-s + 1.39·73-s + 2.05·77-s + 1.50·79-s − 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(3.756679091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.756679091\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.23e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 3.56e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 8.12e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.44e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.43e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.23e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 1.26e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 9.41e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 7.80e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 4.06e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.98e11T + 1.55e24T^{2} \) |
| 43 | \( 1 + 1.91e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 5.51e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 1.20e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 6.58e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 4.21e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 5.45e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.17e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.31e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.57e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.45e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.21e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.01e15T + 6.33e29T^{2} \) |
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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
−13.55029946239815154131491182962, −11.71042564855711020276997227885, −10.84718658438640798765292514496, −9.329877850396626835141623939214, −8.352969839839372752832880214763, −6.59385239426815554142829548737, −5.44988723586899373106480392763, −4.04043341590312171774093887021, −1.95608101380608569342613760921, −1.28364364289480479407578843660,
1.28364364289480479407578843660, 1.95608101380608569342613760921, 4.04043341590312171774093887021, 5.44988723586899373106480392763, 6.59385239426815554142829548737, 8.352969839839372752832880214763, 9.329877850396626835141623939214, 10.84718658438640798765292514496, 11.71042564855711020276997227885, 13.55029946239815154131491182962
