| L(s) = 1 | − 9·3-s − 106.·5-s + 197.·7-s + 81·9-s − 593.·11-s + 999.·13-s + 954.·15-s + 128.·17-s + 1.13e3·19-s − 1.77e3·21-s + 529·23-s + 8.12e3·25-s − 729·27-s + 3.13e3·29-s − 73.9·31-s + 5.34e3·33-s − 2.09e4·35-s − 1.27e4·37-s − 8.99e3·39-s + 4.94e3·41-s − 2.07e4·43-s − 8.59e3·45-s + 1.57e4·47-s + 2.20e4·49-s − 1.15e3·51-s − 4.04e4·53-s + 6.29e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.89·5-s + 1.52·7-s + 0.333·9-s − 1.47·11-s + 1.63·13-s + 1.09·15-s + 0.107·17-s + 0.719·19-s − 0.878·21-s + 0.208·23-s + 2.60·25-s − 0.192·27-s + 0.692·29-s − 0.0138·31-s + 0.853·33-s − 2.88·35-s − 1.52·37-s − 0.946·39-s + 0.459·41-s − 1.71·43-s − 0.632·45-s + 1.03·47-s + 1.31·49-s − 0.0620·51-s − 1.97·53-s + 2.80·55-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
| \(L(3)\) |
\(\approx\) |
\(1.262461289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262461289\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 106.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 197.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 593.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 999.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 128.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.13e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 73.9T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.27e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.08e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.14e5T + 8.58e9T^{2} \) |
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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
−8.661359344675713177231740614229, −8.205194135816580075193223820186, −7.66452382069110581660512557627, −6.82539089456211038534240488936, −5.42882973931276291669920190474, −4.86476460249577778746210954095, −3.99162007498150579134006314809, −3.08617171653022690858109835830, −1.46224252560401867684353414475, −0.53628506571725367382292938792,
0.53628506571725367382292938792, 1.46224252560401867684353414475, 3.08617171653022690858109835830, 3.99162007498150579134006314809, 4.86476460249577778746210954095, 5.42882973931276291669920190474, 6.82539089456211038534240488936, 7.66452382069110581660512557627, 8.205194135816580075193223820186, 8.661359344675713177231740614229
