L(s) = 1 | + 3-s − 2·4-s + 2.44·5-s − 4.44·7-s + 9-s − 2·12-s + 13-s + 2.44·15-s + 4·16-s − 1.44·17-s − 19-s − 4.89·20-s − 4.44·21-s − 23-s + 0.999·25-s + 27-s + 8.89·28-s + 29-s − 10.4·31-s − 10.8·35-s − 2·36-s + 9.89·37-s + 39-s − 9.34·41-s − 0.101·43-s + 2.44·45-s − 12.4·47-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 1.09·5-s − 1.68·7-s + 0.333·9-s − 0.577·12-s + 0.277·13-s + 0.632·15-s + 16-s − 0.351·17-s − 0.229·19-s − 1.09·20-s − 0.970·21-s − 0.208·23-s + 0.199·25-s + 0.192·27-s + 1.68·28-s + 0.185·29-s − 1.87·31-s − 1.84·35-s − 0.333·36-s + 1.62·37-s + 0.160·39-s − 1.45·41-s − 0.0154·43-s + 0.365·45-s − 1.81·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 9.89T + 37T^{2} \) |
| 41 | \( 1 + 9.34T + 41T^{2} \) |
| 43 | \( 1 + 0.101T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 0.550T + 59T^{2} \) |
| 61 | \( 1 + 3.10T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−9.104795992674507996002777626039, −8.209453235055108870077734210692, −7.15386939190466272148671191760, −6.23736341909999860013623991370, −5.73980940392516242861162408648, −4.61346084806988000078946641270, −3.63093431891875533086220189659, −2.94010847927804657570018077431, −1.68640674939053920681755660529, 0,
1.68640674939053920681755660529, 2.94010847927804657570018077431, 3.63093431891875533086220189659, 4.61346084806988000078946641270, 5.73980940392516242861162408648, 6.23736341909999860013623991370, 7.15386939190466272148671191760, 8.209453235055108870077734210692, 9.104795992674507996002777626039