| L(s) = 1 | − 1.07·3-s − 15.2·5-s + 32.3·7-s − 25.8·9-s + 38.9·11-s − 5.91·13-s + 16.4·15-s + 19.9·19-s − 34.8·21-s − 56.2·23-s + 107.·25-s + 56.9·27-s − 236.·29-s − 119.·31-s − 41.9·33-s − 493.·35-s + 133.·37-s + 6.37·39-s + 414.·41-s − 502.·43-s + 394.·45-s − 180.·47-s + 702.·49-s − 100.·53-s − 594.·55-s − 21.4·57-s + 627.·59-s + ⋯ |
| L(s) = 1 | − 0.207·3-s − 1.36·5-s + 1.74·7-s − 0.956·9-s + 1.06·11-s − 0.126·13-s + 0.283·15-s + 0.240·19-s − 0.362·21-s − 0.510·23-s + 0.862·25-s + 0.405·27-s − 1.51·29-s − 0.691·31-s − 0.221·33-s − 2.38·35-s + 0.591·37-s + 0.0261·39-s + 1.57·41-s − 1.78·43-s + 1.30·45-s − 0.559·47-s + 2.04·49-s − 0.261·53-s − 1.45·55-s − 0.0499·57-s + 1.38·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 1.07T + 27T^{2} \) |
| 5 | \( 1 + 15.2T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 11 | \( 1 - 38.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.91T + 2.19e3T^{2} \) |
| 19 | \( 1 - 19.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 56.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 236.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 180.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 100.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 627.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 692.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 573.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 396.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 933.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 278.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 252.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−8.206383078956579084550398836933, −7.67204345814708443771392511924, −6.90085285740286740272860139370, −5.75012330964075276170546967899, −5.04442981640744445781619648396, −4.15363672097523520325054379968, −3.59485877321344864520249896731, −2.21478496402950997261590338908, −1.14382545330230757796393492871, 0,
1.14382545330230757796393492871, 2.21478496402950997261590338908, 3.59485877321344864520249896731, 4.15363672097523520325054379968, 5.04442981640744445781619648396, 5.75012330964075276170546967899, 6.90085285740286740272860139370, 7.67204345814708443771392511924, 8.206383078956579084550398836933
