L(s) = 1 | − 0.750·3-s − 5-s − 4.31·7-s − 2.43·9-s + 1.26·11-s + 0.840·13-s + 0.750·15-s + 1.74·17-s + 4.53·19-s + 3.23·21-s − 7.00·23-s + 25-s + 4.07·27-s − 5.86·29-s + 5.85·31-s − 0.947·33-s + 4.31·35-s + 9.40·37-s − 0.630·39-s − 6.00·41-s + 6.28·43-s + 2.43·45-s + 1.12·47-s + 11.6·49-s − 1.30·51-s + 0.740·53-s − 1.26·55-s + ⋯ |
L(s) = 1 | − 0.433·3-s − 0.447·5-s − 1.63·7-s − 0.812·9-s + 0.380·11-s + 0.233·13-s + 0.193·15-s + 0.422·17-s + 1.03·19-s + 0.706·21-s − 1.46·23-s + 0.200·25-s + 0.784·27-s − 1.08·29-s + 1.05·31-s − 0.164·33-s + 0.729·35-s + 1.54·37-s − 0.100·39-s − 0.937·41-s + 0.958·43-s + 0.363·45-s + 0.163·47-s + 1.66·49-s − 0.183·51-s + 0.101·53-s − 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 0.750T + 3T^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 - 0.840T + 13T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 + 7.00T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 - 5.85T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 + 6.00T + 41T^{2} \) |
| 43 | \( 1 - 6.28T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 - 0.740T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 6.59T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 - 4.74T + 97T^{2} \) |
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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
−7.48414072943001703246911133219, −6.65201583154067798944540203872, −6.01504288857951917354909488534, −5.72428899601038172802727901324, −4.63340429179166672014745363726, −3.71982123321061623275838473044, −3.25123948438750637451580664481, −2.42128268293822831676940906339, −0.939350364232261011605637152371, 0,
0.939350364232261011605637152371, 2.42128268293822831676940906339, 3.25123948438750637451580664481, 3.71982123321061623275838473044, 4.63340429179166672014745363726, 5.72428899601038172802727901324, 6.01504288857951917354909488534, 6.65201583154067798944540203872, 7.48414072943001703246911133219