L(s) = 1 | − 2-s + 4-s + 2.03·5-s − 8-s − 2.03·10-s + 2.87·11-s + 0.617·13-s + 16-s + 0.796·17-s − 8.30·19-s + 2.03·20-s − 2.87·22-s − 1.12·23-s − 0.872·25-s − 0.617·26-s − 6.74·29-s + 4.24·31-s − 32-s − 0.796·34-s + 37-s + 8.30·38-s − 2.03·40-s + 1.41·41-s + 2·43-s + 2.87·44-s + 1.12·46-s + 2.64·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.908·5-s − 0.353·8-s − 0.642·10-s + 0.866·11-s + 0.171·13-s + 0.250·16-s + 0.193·17-s − 1.90·19-s + 0.454·20-s − 0.612·22-s − 0.234·23-s − 0.174·25-s − 0.121·26-s − 1.25·29-s + 0.762·31-s − 0.176·32-s − 0.136·34-s + 0.164·37-s + 1.34·38-s − 0.321·40-s + 0.220·41-s + 0.304·43-s + 0.433·44-s + 0.166·46-s + 0.386·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.03T + 5T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 0.617T + 13T^{2} \) |
| 17 | \( 1 - 0.796T + 17T^{2} \) |
| 19 | \( 1 + 8.30T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 8.87T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 8.30T + 97T^{2} \) |
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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
−7.59745324491311127596505339130, −6.71716924116418125785080751356, −6.15684697157858261907154652220, −5.77556826301885431098179876541, −4.60428827981278453007698974802, −3.92383981850007719676798921756, −2.87242957642189291803162435569, −1.97655153392808806536676488816, −1.40989942242551143371468107475, 0,
1.40989942242551143371468107475, 1.97655153392808806536676488816, 2.87242957642189291803162435569, 3.92383981850007719676798921756, 4.60428827981278453007698974802, 5.77556826301885431098179876541, 6.15684697157858261907154652220, 6.71716924116418125785080751356, 7.59745324491311127596505339130