| L(s) = 1 | + 2.01·3-s + 5-s − 0.690·7-s + 1.05·9-s − 4.32·11-s − 3.30·13-s + 2.01·15-s + 5.28·17-s − 7.62·19-s − 1.38·21-s + 1.60·23-s + 25-s − 3.92·27-s − 2.40·29-s + 4.69·31-s − 8.71·33-s − 0.690·35-s + 11.1·37-s − 6.65·39-s − 5.49·41-s − 0.362·43-s + 1.05·45-s − 4.60·47-s − 6.52·49-s + 10.6·51-s + 7.06·53-s − 4.32·55-s + ⋯ |
| L(s) = 1 | + 1.16·3-s + 0.447·5-s − 0.261·7-s + 0.350·9-s − 1.30·11-s − 0.916·13-s + 0.519·15-s + 1.28·17-s − 1.74·19-s − 0.303·21-s + 0.335·23-s + 0.200·25-s − 0.755·27-s − 0.447·29-s + 0.843·31-s − 1.51·33-s − 0.116·35-s + 1.83·37-s − 1.06·39-s − 0.858·41-s − 0.0552·43-s + 0.156·45-s − 0.672·47-s − 0.931·49-s + 1.49·51-s + 0.969·53-s − 0.583·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 7 | \( 1 + 0.690T + 7T^{2} \) |
| 11 | \( 1 + 4.32T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 - 4.69T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 + 0.362T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 7.06T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 1.81T + 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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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
−7.973056239764656994615673083355, −7.41043597696744400803009859864, −6.44112930881613958206566184594, −5.68010069085070594085479509133, −4.89648301428768506833252379989, −4.04572255566288142428293882232, −2.83793332240171616888635804276, −2.73751411428128194726231136376, −1.67111573731761944893101197215, 0,
1.67111573731761944893101197215, 2.73751411428128194726231136376, 2.83793332240171616888635804276, 4.04572255566288142428293882232, 4.89648301428768506833252379989, 5.68010069085070594085479509133, 6.44112930881613958206566184594, 7.41043597696744400803009859864, 7.973056239764656994615673083355
