L(s) = 1 | − 2.68·2-s − 3-s + 5.21·4-s + 2.43·5-s + 2.68·6-s − 7-s − 8.62·8-s + 9-s − 6.53·10-s − 0.147·11-s − 5.21·12-s + 3.06·13-s + 2.68·14-s − 2.43·15-s + 12.7·16-s + 3.88·17-s − 2.68·18-s + 2.57·19-s + 12.6·20-s + 21-s + 0.395·22-s + 4.81·23-s + 8.62·24-s + 0.922·25-s − 8.22·26-s − 27-s − 5.21·28-s + ⋯ |
L(s) = 1 | − 1.89·2-s − 0.577·3-s + 2.60·4-s + 1.08·5-s + 1.09·6-s − 0.377·7-s − 3.04·8-s + 0.333·9-s − 2.06·10-s − 0.0443·11-s − 1.50·12-s + 0.849·13-s + 0.717·14-s − 0.628·15-s + 3.18·16-s + 0.943·17-s − 0.632·18-s + 0.589·19-s + 2.83·20-s + 0.218·21-s + 0.0842·22-s + 1.00·23-s + 1.76·24-s + 0.184·25-s − 1.61·26-s − 0.192·27-s − 0.984·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 11 | \( 1 + 0.147T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 - 0.103T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 7.89T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 - 0.609T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 + 0.516T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 4.28T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−7.52060525541786896697192816143, −6.84793718793295870318328796249, −6.38276139490847562221960079655, −5.70264659147715950101663910998, −5.08824134965176104708058726187, −3.45972621555858546379541599932, −2.82472138904873791247097839098, −1.59974695369257422376108725751, −1.28837266621685142774755196471, 0,
1.28837266621685142774755196471, 1.59974695369257422376108725751, 2.82472138904873791247097839098, 3.45972621555858546379541599932, 5.08824134965176104708058726187, 5.70264659147715950101663910998, 6.38276139490847562221960079655, 6.84793718793295870318328796249, 7.52060525541786896697192816143