L(s) = 1 | + 2.32·3-s − 5-s − 2.44·7-s + 2.40·9-s − 5.19·11-s + 2.45·13-s − 2.32·15-s + 3.30·17-s + 7.37·19-s − 5.69·21-s − 2.41·23-s + 25-s − 1.38·27-s − 0.883·29-s − 0.774·31-s − 12.0·33-s + 2.44·35-s − 3.11·37-s + 5.70·39-s + 11.0·41-s − 7.16·43-s − 2.40·45-s + 5.67·47-s − 1.00·49-s + 7.68·51-s − 10.4·53-s + 5.19·55-s + ⋯ |
L(s) = 1 | + 1.34·3-s − 0.447·5-s − 0.925·7-s + 0.801·9-s − 1.56·11-s + 0.681·13-s − 0.600·15-s + 0.801·17-s + 1.69·19-s − 1.24·21-s − 0.503·23-s + 0.200·25-s − 0.266·27-s − 0.163·29-s − 0.139·31-s − 2.10·33-s + 0.413·35-s − 0.512·37-s + 0.914·39-s + 1.73·41-s − 1.09·43-s − 0.358·45-s + 0.827·47-s − 0.143·49-s + 1.07·51-s − 1.43·53-s + 0.701·55-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 | \( 1 - 2.32T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 - 3.30T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 + 0.883T + 29T^{2} \) |
| 31 | \( 1 + 0.774T + 31T^{2} \) |
| 37 | \( 1 + 3.11T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 8.65T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 7.03T + 67T^{2} \) |
| 71 | \( 1 + 7.87T + 71T^{2} \) |
| 73 | \( 1 - 8.30T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 4.17T + 83T^{2} \) |
| 89 | \( 1 + 9.39T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−7.66401842269759020857920356137, −7.15190198024425639411216949175, −6.01755874742692749824692317526, −5.47985015365171362876247468970, −4.50238231741634672288939128037, −3.47745466639925396081563120565, −3.18726332929609965392041333046, −2.57520450038383011255849751028, −1.38635712212457435354054933021, 0,
1.38635712212457435354054933021, 2.57520450038383011255849751028, 3.18726332929609965392041333046, 3.47745466639925396081563120565, 4.50238231741634672288939128037, 5.47985015365171362876247468970, 6.01755874742692749824692317526, 7.15190198024425639411216949175, 7.66401842269759020857920356137