| L(s) = 1 | + 3·3-s − 33.1·7-s + 9·9-s + 63.9·11-s + 89.7·13-s + 71.4·17-s − 44.1·19-s − 99.4·21-s − 100·23-s + 27·27-s + 33.4·29-s + 120.·31-s + 191.·33-s + 13.5·37-s + 269.·39-s + 130.·41-s − 410.·43-s − 216.·47-s + 755.·49-s + 214.·51-s − 279.·53-s − 132.·57-s − 133.·59-s − 297.·61-s − 298.·63-s + 501.·67-s − 300·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·7-s + 0.333·9-s + 1.75·11-s + 1.91·13-s + 1.01·17-s − 0.533·19-s − 1.03·21-s − 0.906·23-s + 0.192·27-s + 0.213·29-s + 0.697·31-s + 1.01·33-s + 0.0600·37-s + 1.10·39-s + 0.497·41-s − 1.45·43-s − 0.671·47-s + 2.20·49-s + 0.588·51-s − 0.725·53-s − 0.307·57-s − 0.294·59-s − 0.624·61-s − 0.596·63-s + 0.913·67-s − 0.523·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.870104806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.870104806\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 33.1T + 343T^{2} \) |
| 11 | \( 1 - 63.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 13.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 410.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 216.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 279.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 133.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 297.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 501.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 289.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 756.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 120.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 65.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 828.T + 9.12e5T^{2} \) |
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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
−8.649066918136683265331342679800, −8.059878974106039676551409041768, −6.75008300886041655096346964511, −6.44153460362626432737443692358, −5.83111042970127777990547085930, −4.23226841207793358487634758069, −3.58217253574633856493256609173, −3.16331733616599652013292737268, −1.70595193877541310289151756001, −0.76295764350744276908756855379,
0.76295764350744276908756855379, 1.70595193877541310289151756001, 3.16331733616599652013292737268, 3.58217253574633856493256609173, 4.23226841207793358487634758069, 5.83111042970127777990547085930, 6.44153460362626432737443692358, 6.75008300886041655096346964511, 8.059878974106039676551409041768, 8.649066918136683265331342679800
