| L(s) = 1 | + 4.30·3-s − 8.06·5-s − 26.0·7-s − 8.43·9-s + 3.26·13-s − 34.7·15-s + 20.8·17-s − 125.·19-s − 112.·21-s − 97.8·23-s − 60.0·25-s − 152.·27-s + 263.·29-s − 199.·31-s + 210.·35-s + 365.·37-s + 14.0·39-s − 273.·41-s + 388.·43-s + 67.9·45-s + 51.8·47-s + 337.·49-s + 89.9·51-s − 412.·53-s − 542.·57-s − 26.2·59-s − 164.·61-s + ⋯ |
| L(s) = 1 | + 0.829·3-s − 0.721·5-s − 1.40·7-s − 0.312·9-s + 0.0696·13-s − 0.597·15-s + 0.297·17-s − 1.52·19-s − 1.16·21-s − 0.886·23-s − 0.480·25-s − 1.08·27-s + 1.68·29-s − 1.15·31-s + 1.01·35-s + 1.62·37-s + 0.0577·39-s − 1.04·41-s + 1.37·43-s + 0.225·45-s + 0.160·47-s + 0.982·49-s + 0.246·51-s − 1.06·53-s − 1.26·57-s − 0.0579·59-s − 0.344·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.019930073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019930073\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 4.30T + 27T^{2} \) |
| 5 | \( 1 + 8.06T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 13 | \( 1 - 3.26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 51.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 412.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 164.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 516.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 241.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 273.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 72.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−8.767703690758766245714274342454, −8.132738931271983610469065635034, −7.44354986882135623142132688028, −6.39816508811314803001381256015, −5.91813534139658152060722270573, −4.45109813338907473797905924811, −3.71167311552325241283921011025, −3.01020573132939569888964644344, −2.13867120489291702443565879740, −0.41956349058395375079334595958,
0.41956349058395375079334595958, 2.13867120489291702443565879740, 3.01020573132939569888964644344, 3.71167311552325241283921011025, 4.45109813338907473797905924811, 5.91813534139658152060722270573, 6.39816508811314803001381256015, 7.44354986882135623142132688028, 8.132738931271983610469065635034, 8.767703690758766245714274342454
