| L(s) = 1 | + 7.68·3-s + 15.7·5-s + 19.8·7-s + 32.0·9-s + 11·11-s − 13·13-s + 120.·15-s + 7.57·17-s − 1.78·19-s + 152.·21-s − 6.71·23-s + 121.·25-s + 39.0·27-s + 85.8·29-s − 135.·31-s + 84.5·33-s + 312.·35-s − 333.·37-s − 99.9·39-s + 17.7·41-s − 106.·43-s + 503.·45-s + 92.5·47-s + 51.8·49-s + 58.2·51-s − 244.·53-s + 172.·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 1.40·5-s + 1.07·7-s + 1.18·9-s + 0.301·11-s − 0.277·13-s + 2.07·15-s + 0.108·17-s − 0.0215·19-s + 1.58·21-s − 0.0608·23-s + 0.974·25-s + 0.278·27-s + 0.549·29-s − 0.784·31-s + 0.445·33-s + 1.50·35-s − 1.48·37-s − 0.410·39-s + 0.0677·41-s − 0.377·43-s + 1.66·45-s + 0.287·47-s + 0.151·49-s + 0.159·51-s − 0.633·53-s + 0.423·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.875002314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.875002314\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 7.68T + 27T^{2} \) |
| 5 | \( 1 - 15.7T + 125T^{2} \) |
| 7 | \( 1 - 19.8T + 343T^{2} \) |
| 17 | \( 1 - 7.57T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.78T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.71T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 92.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 244.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 792.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 423.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 543.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 465.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 102.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 55.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 649.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 141.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 912.T + 9.12e5T^{2} \) |
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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
−10.06232573638346338379634477098, −9.339298744594617627154938090804, −8.661370498009900731120759316535, −7.88040182372089375475037719681, −6.87519404329366735230874457463, −5.65764296844575329293227230806, −4.66402147626118778785467521925, −3.33981906219135146903986113953, −2.18742202928476696523509778944, −1.55645986058097202722516663971,
1.55645986058097202722516663971, 2.18742202928476696523509778944, 3.33981906219135146903986113953, 4.66402147626118778785467521925, 5.65764296844575329293227230806, 6.87519404329366735230874457463, 7.88040182372089375475037719681, 8.661370498009900731120759316535, 9.339298744594617627154938090804, 10.06232573638346338379634477098
