| L(s) = 1 | + 1.50·3-s − 0.778·5-s − 16.0·7-s − 24.7·9-s + 7.86·11-s + 64.3·13-s − 1.17·15-s + 9.88·19-s − 24.0·21-s + 190.·23-s − 124.·25-s − 77.9·27-s − 210.·29-s − 6.86·31-s + 11.8·33-s + 12.4·35-s − 92.4·37-s + 96.9·39-s + 318.·41-s + 227.·43-s + 19.2·45-s − 240.·47-s − 86.9·49-s + 247.·53-s − 6.12·55-s + 14.8·57-s − 321.·59-s + ⋯ |
| L(s) = 1 | + 0.289·3-s − 0.0696·5-s − 0.863·7-s − 0.916·9-s + 0.215·11-s + 1.37·13-s − 0.0201·15-s + 0.119·19-s − 0.250·21-s + 1.72·23-s − 0.995·25-s − 0.555·27-s − 1.34·29-s − 0.0397·31-s + 0.0624·33-s + 0.0601·35-s − 0.410·37-s + 0.398·39-s + 1.21·41-s + 0.807·43-s + 0.0638·45-s − 0.747·47-s − 0.253·49-s + 0.642·53-s − 0.0150·55-s + 0.0345·57-s − 0.709·59-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 | \( 1 - 1.50T + 27T^{2} \) |
| 5 | \( 1 + 0.778T + 125T^{2} \) |
| 7 | \( 1 + 16.0T + 343T^{2} \) |
| 11 | \( 1 - 7.86T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 9.88T + 6.85e3T^{2} \) |
| 23 | \( 1 - 190.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 210.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 6.86T + 2.97e4T^{2} \) |
| 37 | \( 1 + 92.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 227.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 443.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 17.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 575.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 490.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 187.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 58.9T + 9.12e5T^{2} \) |
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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
−8.332177921040554223270514725025, −7.56503206034152372670723071502, −6.62209800370261178678702152259, −5.96740817444666596312526679147, −5.25501432031204484214825156007, −3.90017701869678417676667763766, −3.38975625709537815328854647314, −2.47398500579397701938243449557, −1.18282962955843038793958361529, 0,
1.18282962955843038793958361529, 2.47398500579397701938243449557, 3.38975625709537815328854647314, 3.90017701869678417676667763766, 5.25501432031204484214825156007, 5.96740817444666596312526679147, 6.62209800370261178678702152259, 7.56503206034152372670723071502, 8.332177921040554223270514725025
