| L(s) = 1 | + 2.52·3-s − 5-s − 4.03·7-s + 3.39·9-s + 1.67·11-s + 4.79·13-s − 2.52·15-s − 3.28·17-s − 19-s − 10.2·21-s − 4.03·23-s + 25-s + 0.988·27-s − 1.14·29-s − 9.35·31-s + 4.23·33-s + 4.03·35-s + 4.43·37-s + 12.1·39-s + 3.21·41-s + 4.89·43-s − 3.39·45-s − 6.65·47-s + 9.30·49-s − 8.30·51-s + 0.224·53-s − 1.67·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s − 0.447·5-s − 1.52·7-s + 1.13·9-s + 0.505·11-s + 1.32·13-s − 0.652·15-s − 0.796·17-s − 0.229·19-s − 2.22·21-s − 0.841·23-s + 0.200·25-s + 0.190·27-s − 0.212·29-s − 1.67·31-s + 0.737·33-s + 0.682·35-s + 0.728·37-s + 1.94·39-s + 0.502·41-s + 0.746·43-s − 0.505·45-s − 0.970·47-s + 1.32·49-s − 1.16·51-s + 0.0308·53-s − 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 9.35T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 - 0.224T + 53T^{2} \) |
| 59 | \( 1 - 5.52T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 + 4.98T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−7.72439857712141411292403630548, −7.19239376433992938585277680572, −6.29675972334452097881277046879, −5.90136256668388925194816398408, −4.36008552596490965105025896843, −3.78955106248052868805425185798, −3.32576948442519179525523618890, −2.53561718008445474249134133680, −1.53371405299089746699836874318, 0,
1.53371405299089746699836874318, 2.53561718008445474249134133680, 3.32576948442519179525523618890, 3.78955106248052868805425185798, 4.36008552596490965105025896843, 5.90136256668388925194816398408, 6.29675972334452097881277046879, 7.19239376433992938585277680572, 7.72439857712141411292403630548
