| L(s) = 1 | − 0.378·2-s − 1.85·4-s + 3.79·5-s + 4.13·7-s + 1.45·8-s − 1.43·10-s − 6.25·11-s − 2.85·13-s − 1.56·14-s + 3.16·16-s + 0.304·17-s − 0.261·19-s − 7.03·20-s + 2.36·22-s − 6.24·23-s + 9.36·25-s + 1.08·26-s − 7.67·28-s + 3.98·29-s + 1.52·31-s − 4.11·32-s − 0.115·34-s + 15.6·35-s + 6.41·37-s + 0.0987·38-s + 5.52·40-s + 4.56·41-s + ⋯ |
| L(s) = 1 | − 0.267·2-s − 0.928·4-s + 1.69·5-s + 1.56·7-s + 0.515·8-s − 0.453·10-s − 1.88·11-s − 0.792·13-s − 0.417·14-s + 0.790·16-s + 0.0739·17-s − 0.0599·19-s − 1.57·20-s + 0.504·22-s − 1.30·23-s + 1.87·25-s + 0.211·26-s − 1.44·28-s + 0.740·29-s + 0.273·31-s − 0.726·32-s − 0.0197·34-s + 2.64·35-s + 1.05·37-s + 0.0160·38-s + 0.873·40-s + 0.712·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.926703217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.926703217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.378T + 2T^{2} \) |
| 5 | \( 1 - 3.79T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 + 6.25T + 11T^{2} \) |
| 13 | \( 1 + 2.85T + 13T^{2} \) |
| 17 | \( 1 - 0.304T + 17T^{2} \) |
| 19 | \( 1 + 0.261T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 6.41T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 6.15T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 5.78T + 61T^{2} \) |
| 67 | \( 1 + 0.724T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 4.57T + 83T^{2} \) |
| 89 | \( 1 + 7.83T + 89T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−8.584151396922532905045926459968, −7.917874695342754285072303438106, −7.46412445756046948739935685962, −6.04021458395783358783522108494, −5.42048675748869545600721605977, −4.97141760648301025853580211613, −4.26841448084167008047846200529, −2.55195831346728254101913535737, −2.11332719682665898815380730617, −0.880553600694343690723401781701,
0.880553600694343690723401781701, 2.11332719682665898815380730617, 2.55195831346728254101913535737, 4.26841448084167008047846200529, 4.97141760648301025853580211613, 5.42048675748869545600721605977, 6.04021458395783358783522108494, 7.46412445756046948739935685962, 7.917874695342754285072303438106, 8.584151396922532905045926459968
