| L(s) = 1 | − 2.73·2-s + 5.45·4-s + 0.936·5-s + 0.519·7-s − 9.43·8-s − 2.55·10-s + 4.70·13-s − 1.41·14-s + 14.8·16-s − 2.69·17-s − 3.41·19-s + 5.10·20-s − 6.97·23-s − 4.12·25-s − 12.8·26-s + 2.83·28-s + 4.18·29-s + 5.18·31-s − 21.6·32-s + 7.35·34-s + 0.486·35-s + 2.06·37-s + 9.33·38-s − 8.83·40-s + 0.173·41-s + 2.26·43-s + 19.0·46-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.72·4-s + 0.418·5-s + 0.196·7-s − 3.33·8-s − 0.808·10-s + 1.30·13-s − 0.378·14-s + 3.71·16-s − 0.652·17-s − 0.784·19-s + 1.14·20-s − 1.45·23-s − 0.824·25-s − 2.51·26-s + 0.535·28-s + 0.777·29-s + 0.931·31-s − 3.83·32-s + 1.26·34-s + 0.0821·35-s + 0.340·37-s + 1.51·38-s − 1.39·40-s + 0.0270·41-s + 0.346·43-s + 2.80·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 5 | \( 1 - 0.936T + 5T^{2} \) |
| 7 | \( 1 - 0.519T + 7T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 - 0.173T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 0.307T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 + 3.97T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 2.90T + 71T^{2} \) |
| 73 | \( 1 + 9.52T + 73T^{2} \) |
| 79 | \( 1 - 2.04T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−7.70274434558512235474636181026, −6.66384410044719552784208823091, −6.25079000736481497438751702211, −5.82894738307797173377737893385, −4.49339529147414993810066608931, −3.54665427173236086779875513258, −2.51757341938885887519181283610, −1.90028912445806282713051467689, −1.12801753451184782356057992812, 0,
1.12801753451184782356057992812, 1.90028912445806282713051467689, 2.51757341938885887519181283610, 3.54665427173236086779875513258, 4.49339529147414993810066608931, 5.82894738307797173377737893385, 6.25079000736481497438751702211, 6.66384410044719552784208823091, 7.70274434558512235474636181026
