| L(s) = 1 | − 1.73·3-s + 1.63·5-s + 7-s + 0.00974·9-s − 11-s − 13-s − 2.84·15-s + 4.66·17-s + 1.13·19-s − 1.73·21-s − 6.21·23-s − 2.31·25-s + 5.18·27-s − 5.84·29-s + 4.87·31-s + 1.73·33-s + 1.63·35-s − 8.62·37-s + 1.73·39-s − 3.59·41-s + 1.28·43-s + 0.0159·45-s + 7.07·47-s + 49-s − 8.09·51-s + 0.181·53-s − 1.63·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.732·5-s + 0.377·7-s + 0.00324·9-s − 0.301·11-s − 0.277·13-s − 0.733·15-s + 1.13·17-s + 0.261·19-s − 0.378·21-s − 1.29·23-s − 0.463·25-s + 0.998·27-s − 1.08·29-s + 0.875·31-s + 0.302·33-s + 0.276·35-s − 1.41·37-s + 0.277·39-s − 0.561·41-s + 0.196·43-s + 0.00238·45-s + 1.03·47-s + 0.142·49-s − 1.13·51-s + 0.0248·53-s − 0.220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 17 | \( 1 - 4.66T + 17T^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 + 5.84T + 29T^{2} \) |
| 31 | \( 1 - 4.87T + 31T^{2} \) |
| 37 | \( 1 + 8.62T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 - 1.28T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 0.181T + 53T^{2} \) |
| 59 | \( 1 - 0.206T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 3.09T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 + 7.63T + 83T^{2} \) |
| 89 | \( 1 - 0.284T + 89T^{2} \) |
| 97 | \( 1 - 0.615T + 97T^{2} \) |
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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
−7.993065339585768296634069870876, −7.33608312029496811520038939754, −6.39478062929236968301487296155, −5.64474834323764593819350385732, −5.43891241553240768583015802820, −4.48323627906219234657414348249, −3.42865592252877289986919373698, −2.32438818691717494668125752618, −1.35718394343569597956690082645, 0,
1.35718394343569597956690082645, 2.32438818691717494668125752618, 3.42865592252877289986919373698, 4.48323627906219234657414348249, 5.43891241553240768583015802820, 5.64474834323764593819350385732, 6.39478062929236968301487296155, 7.33608312029496811520038939754, 7.993065339585768296634069870876
