| L(s) = 1 | + 0.141·3-s + 0.343·5-s − 7-s − 2.97·9-s + 11-s + 13-s + 0.0486·15-s − 1.94·17-s − 0.383·19-s − 0.141·21-s − 1.66·23-s − 4.88·25-s − 0.846·27-s + 3.52·29-s + 8.92·31-s + 0.141·33-s − 0.343·35-s + 5.87·37-s + 0.141·39-s + 6.78·41-s − 1.87·43-s − 1.02·45-s + 2.51·47-s + 49-s − 0.275·51-s + 0.793·53-s + 0.343·55-s + ⋯ |
| L(s) = 1 | + 0.0817·3-s + 0.153·5-s − 0.377·7-s − 0.993·9-s + 0.301·11-s + 0.277·13-s + 0.0125·15-s − 0.471·17-s − 0.0880·19-s − 0.0308·21-s − 0.347·23-s − 0.976·25-s − 0.162·27-s + 0.655·29-s + 1.60·31-s + 0.0246·33-s − 0.0581·35-s + 0.965·37-s + 0.0226·39-s + 1.05·41-s − 0.286·43-s − 0.152·45-s + 0.366·47-s + 0.142·49-s − 0.0385·51-s + 0.108·53-s + 0.0463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.141T + 3T^{2} \) |
| 5 | \( 1 - 0.343T + 5T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 19 | \( 1 + 0.383T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 - 6.78T + 41T^{2} \) |
| 43 | \( 1 + 1.87T + 43T^{2} \) |
| 47 | \( 1 - 2.51T + 47T^{2} \) |
| 53 | \( 1 - 0.793T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 6.36T + 61T^{2} \) |
| 67 | \( 1 + 8.05T + 67T^{2} \) |
| 71 | \( 1 - 0.428T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 9.37T + 79T^{2} \) |
| 83 | \( 1 + 4.37T + 83T^{2} \) |
| 89 | \( 1 + 4.95T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.61850407614653280347100567089, −6.58103652266934672820967762012, −6.16433392930943096906685808939, −5.57413645180659260664473490946, −4.55882770411562545675219876313, −3.95510625694091823356048027662, −2.94321831668259577596440716076, −2.43553338247583224016173300875, −1.22335425551692589712406716149, 0,
1.22335425551692589712406716149, 2.43553338247583224016173300875, 2.94321831668259577596440716076, 3.95510625694091823356048027662, 4.55882770411562545675219876313, 5.57413645180659260664473490946, 6.16433392930943096906685808939, 6.58103652266934672820967762012, 7.61850407614653280347100567089
