| L(s) = 1 | − 2-s − 3-s + 4-s + 1.72·5-s + 6-s + 2.39·7-s − 8-s + 9-s − 1.72·10-s − 4.25·11-s − 12-s + 13-s − 2.39·14-s − 1.72·15-s + 16-s + 6.64·17-s − 18-s + 1.39·19-s + 1.72·20-s − 2.39·21-s + 4.25·22-s + 0.660·23-s + 24-s − 2.03·25-s − 26-s − 27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.769·5-s + 0.408·6-s + 0.903·7-s − 0.353·8-s + 0.333·9-s − 0.544·10-s − 1.28·11-s − 0.288·12-s + 0.277·13-s − 0.638·14-s − 0.444·15-s + 0.250·16-s + 1.61·17-s − 0.235·18-s + 0.318·19-s + 0.384·20-s − 0.521·21-s + 0.906·22-s + 0.137·23-s + 0.204·24-s − 0.407·25-s − 0.196·26-s − 0.192·27-s + 0.451·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 1.72T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 - 0.660T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 - 0.542T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 + 5.06T + 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 - 0.644T + 59T^{2} \) |
| 61 | \( 1 - 7.01T + 61T^{2} \) |
| 67 | \( 1 + 8.17T + 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 + 1.21T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.66667656221158738613175457345, −6.95251978835286772283597707778, −5.91273149018515663826851653407, −5.46678202843149369370223826114, −5.08177966246653373125682033532, −3.84697083873084914402180276043, −2.91116779127382566635657858161, −1.88550776225120894918549782352, −1.32894695600967012567916634250, 0,
1.32894695600967012567916634250, 1.88550776225120894918549782352, 2.91116779127382566635657858161, 3.84697083873084914402180276043, 5.08177966246653373125682033532, 5.46678202843149369370223826114, 5.91273149018515663826851653407, 6.95251978835286772283597707778, 7.66667656221158738613175457345
