L(s) = 1 | + 3.11·3-s − 5-s − 4.50·7-s + 6.72·9-s − 4.33·11-s − 3.72·13-s − 3.11·15-s + 1.11·17-s − 4.50·19-s − 14.0·21-s + 23-s + 25-s + 11.6·27-s − 8.23·29-s − 1.72·31-s − 13.5·33-s + 4.50·35-s − 0.781·37-s − 11.6·39-s + 3.90·41-s − 8·43-s − 6.72·45-s + 11.4·47-s + 13.3·49-s + 3.49·51-s − 6·53-s + 4.33·55-s + ⋯ |
L(s) = 1 | + 1.80·3-s − 0.447·5-s − 1.70·7-s + 2.24·9-s − 1.30·11-s − 1.03·13-s − 0.805·15-s + 0.271·17-s − 1.03·19-s − 3.06·21-s + 0.208·23-s + 0.200·25-s + 2.23·27-s − 1.52·29-s − 0.310·31-s − 2.35·33-s + 0.762·35-s − 0.128·37-s − 1.86·39-s + 0.609·41-s − 1.21·43-s − 1.00·45-s + 1.67·47-s + 1.90·49-s + 0.488·51-s − 0.824·53-s + 0.584·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 3.11T + 3T^{2} \) |
| 7 | \( 1 + 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 + 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 2.43T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 + 0.642T + 97T^{2} \) |
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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
−8.922404387180543480413678521709, −8.067792522882624271632043906777, −7.41031869553070581474697889156, −6.87380197800133543890469895292, −5.63101603209270277629491108185, −4.38710566350799376266440385759, −3.53261079916272994701441132978, −2.86783702482405152928881422469, −2.18143918793632073363691886717, 0,
2.18143918793632073363691886717, 2.86783702482405152928881422469, 3.53261079916272994701441132978, 4.38710566350799376266440385759, 5.63101603209270277629491108185, 6.87380197800133543890469895292, 7.41031869553070581474697889156, 8.067792522882624271632043906777, 8.922404387180543480413678521709