| L(s) = 1 | − 9.24·3-s − 0.656·5-s − 6.14·7-s + 58.4·9-s − 65.3·11-s + 49.7·13-s + 6.07·15-s + 55.4·17-s − 64.7·19-s + 56.7·21-s − 93.8·23-s − 124.·25-s − 290.·27-s − 29·29-s + 236.·31-s + 603.·33-s + 4.03·35-s − 76.8·37-s − 460.·39-s + 215.·41-s + 80.8·43-s − 38.3·45-s + 357.·47-s − 305.·49-s − 512.·51-s − 328.·53-s + 42.9·55-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 0.0587·5-s − 0.331·7-s + 2.16·9-s − 1.79·11-s + 1.06·13-s + 0.104·15-s + 0.791·17-s − 0.781·19-s + 0.589·21-s − 0.851·23-s − 0.996·25-s − 2.07·27-s − 0.185·29-s + 1.36·31-s + 3.18·33-s + 0.0194·35-s − 0.341·37-s − 1.88·39-s + 0.819·41-s + 0.286·43-s − 0.127·45-s + 1.11·47-s − 0.890·49-s − 1.40·51-s − 0.851·53-s + 0.105·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 9.24T + 27T^{2} \) |
| 5 | \( 1 + 0.656T + 125T^{2} \) |
| 7 | \( 1 + 6.14T + 343T^{2} \) |
| 11 | \( 1 + 65.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 93.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 76.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 80.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 844.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 378.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 353.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 696.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 805.T + 9.12e5T^{2} \) |
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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
−8.223038928852274976869611244805, −7.69516057608742298548877800533, −6.60517051159952172035133723421, −5.96982275488915623468659792517, −5.45298681284297335728083161514, −4.59136379567426418606593711008, −3.64454530397629233164011456716, −2.23523964649658084731128072608, −0.878947897967591825975011250940, 0,
0.878947897967591825975011250940, 2.23523964649658084731128072608, 3.64454530397629233164011456716, 4.59136379567426418606593711008, 5.45298681284297335728083161514, 5.96982275488915623468659792517, 6.60517051159952172035133723421, 7.69516057608742298548877800533, 8.223038928852274976869611244805
