| L(s) = 1 | + 2.44·3-s − 3.44·5-s + 7-s + 2.99·9-s − 6.44·11-s − 13-s − 8.44·15-s + 2.44·17-s − 0.550·19-s + 2.44·21-s − 3.89·23-s + 6.89·25-s − 5.89·29-s − 3.44·31-s − 15.7·33-s − 3.44·35-s − 8.44·37-s − 2.44·39-s + 10.8·41-s − 43-s − 10.3·45-s − 3.44·47-s + 49-s + 5.99·51-s + 1.89·53-s + 22.2·55-s − 1.34·57-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 1.54·5-s + 0.377·7-s + 0.999·9-s − 1.94·11-s − 0.277·13-s − 2.18·15-s + 0.594·17-s − 0.126·19-s + 0.534·21-s − 0.812·23-s + 1.37·25-s − 1.09·29-s − 0.619·31-s − 2.75·33-s − 0.583·35-s − 1.38·37-s − 0.392·39-s + 1.70·41-s − 0.152·43-s − 1.54·45-s − 0.503·47-s + 0.142·49-s + 0.840·51-s + 0.260·53-s + 2.99·55-s − 0.178·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 + 0.550T + 19T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 + 3.44T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 1.89T + 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6.89T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 2.10T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.845371129730723967554545584376, −8.132387587166883071479504580285, −7.67409235701984372739636576867, −7.34583102010641700425176161061, −5.64211993005444421707029681347, −4.67039898857974993423596354831, −3.73717950064701044936674308446, −3.04738977331923578819196461222, −2.06816494975524882831877428014, 0,
2.06816494975524882831877428014, 3.04738977331923578819196461222, 3.73717950064701044936674308446, 4.67039898857974993423596354831, 5.64211993005444421707029681347, 7.34583102010641700425176161061, 7.67409235701984372739636576867, 8.132387587166883071479504580285, 8.845371129730723967554545584376
