| L(s) = 1 | − 9·3-s − 12.9·5-s − 52.7·7-s + 81·9-s + 126.·11-s + 817.·13-s + 116.·15-s − 1.41e3·17-s + 2.88e3·19-s + 475.·21-s + 529·23-s − 2.95e3·25-s − 729·27-s + 4.04e3·29-s − 872.·31-s − 1.13e3·33-s + 682.·35-s + 6.60e3·37-s − 7.36e3·39-s + 841.·41-s + 1.75e4·43-s − 1.04e3·45-s − 2.08e4·47-s − 1.40e4·49-s + 1.27e4·51-s − 1.28e4·53-s − 1.63e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.231·5-s − 0.407·7-s + 0.333·9-s + 0.314·11-s + 1.34·13-s + 0.133·15-s − 1.18·17-s + 1.83·19-s + 0.235·21-s + 0.208·23-s − 0.946·25-s − 0.192·27-s + 0.894·29-s − 0.163·31-s − 0.181·33-s + 0.0941·35-s + 0.793·37-s − 0.774·39-s + 0.0781·41-s + 1.44·43-s − 0.0770·45-s − 1.37·47-s − 0.834·49-s + 0.684·51-s − 0.627·53-s − 0.0727·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.669474828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669474828\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 + 12.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 52.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 126.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 817.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.41e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.88e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 872.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 841.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.75e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.08e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.29e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.76e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.40e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.76e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.24e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.17e4T + 8.58e9T^{2} \) |
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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
−9.255098857577908953716459900473, −8.284766110571176049507442087132, −7.39752176367106082078841250672, −6.46165231292784414715090577364, −5.91120525623449700832447365545, −4.81985477283187372186682058199, −3.90337687220320170798463664820, −2.99776126583759331762637733197, −1.54232043949289589763584495955, −0.60172679052712241353395042943,
0.60172679052712241353395042943, 1.54232043949289589763584495955, 2.99776126583759331762637733197, 3.90337687220320170798463664820, 4.81985477283187372186682058199, 5.91120525623449700832447365545, 6.46165231292784414715090577364, 7.39752176367106082078841250672, 8.284766110571176049507442087132, 9.255098857577908953716459900473
