| L(s) = 1 | + 1.16·2-s − 3-s − 0.640·4-s + 5-s − 1.16·6-s − 0.957·7-s − 3.07·8-s + 9-s + 1.16·10-s + 5.05·11-s + 0.640·12-s − 1.11·14-s − 15-s − 2.30·16-s − 1.25·17-s + 1.16·18-s − 2.44·19-s − 0.640·20-s + 0.957·21-s + 5.89·22-s − 5.45·23-s + 3.07·24-s + 25-s − 27-s + 0.613·28-s + 10.2·29-s − 1.16·30-s + ⋯ |
| L(s) = 1 | + 0.824·2-s − 0.577·3-s − 0.320·4-s + 0.447·5-s − 0.475·6-s − 0.361·7-s − 1.08·8-s + 0.333·9-s + 0.368·10-s + 1.52·11-s + 0.184·12-s − 0.298·14-s − 0.258·15-s − 0.576·16-s − 0.304·17-s + 0.274·18-s − 0.561·19-s − 0.143·20-s + 0.208·21-s + 1.25·22-s − 1.13·23-s + 0.628·24-s + 0.200·25-s − 0.192·27-s + 0.115·28-s + 1.89·29-s − 0.212·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.973391453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973391453\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.16T + 2T^{2} \) |
| 7 | \( 1 + 0.957T + 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 - 0.939T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 0.259T + 43T^{2} \) |
| 47 | \( 1 + 0.115T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 0.977T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.930029307788042825073521777146, −8.318041317551867077336258455824, −6.95265387675440704509009359736, −6.27242296174494847191021391777, −5.95760560571474468084036793376, −4.83317119582240771269829038995, −4.24694253457219440892710126379, −3.47740695258340344749618679762, −2.25573195870162832218318980213, −0.821942073501937460199751477560,
0.821942073501937460199751477560, 2.25573195870162832218318980213, 3.47740695258340344749618679762, 4.24694253457219440892710126379, 4.83317119582240771269829038995, 5.95760560571474468084036793376, 6.27242296174494847191021391777, 6.95265387675440704509009359736, 8.318041317551867077336258455824, 8.930029307788042825073521777146
