| L(s) = 1 | + 3-s + 3.16·5-s + 7-s + 9-s + 6.43·11-s + 0.537·13-s + 3.16·15-s − 1.70·17-s − 4.43·19-s + 21-s + 23-s + 5.00·25-s + 27-s + 2.62·29-s + 2.62·31-s + 6.43·33-s + 3.16·35-s + 3.70·37-s + 0.537·39-s − 10.7·41-s − 2.00·43-s + 3.16·45-s + 3.81·47-s + 49-s − 1.70·51-s − 2.82·53-s + 20.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.41·5-s + 0.377·7-s + 0.333·9-s + 1.94·11-s + 0.149·13-s + 0.816·15-s − 0.412·17-s − 1.01·19-s + 0.218·21-s + 0.208·23-s + 1.00·25-s + 0.192·27-s + 0.487·29-s + 0.471·31-s + 1.12·33-s + 0.534·35-s + 0.608·37-s + 0.0861·39-s − 1.68·41-s − 0.306·43-s + 0.471·45-s + 0.555·47-s + 0.142·49-s − 0.238·51-s − 0.387·53-s + 2.74·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.327105465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.327105465\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 - 6.43T + 11T^{2} \) |
| 13 | \( 1 - 0.537T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 3.70T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.42T + 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 6.09T + 97T^{2} \) |
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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
−8.083920722354665600101333824082, −6.80125930225774404579034064904, −6.64170183687900823160057651040, −5.93212520592223804596028593850, −5.01594730667910845480537520293, −4.26733599447819891692850250015, −3.55478801855422007427335837978, −2.47199553108069829720410925321, −1.82061239721307297632322921474, −1.11607938391863845031644846660,
1.11607938391863845031644846660, 1.82061239721307297632322921474, 2.47199553108069829720410925321, 3.55478801855422007427335837978, 4.26733599447819891692850250015, 5.01594730667910845480537520293, 5.93212520592223804596028593850, 6.64170183687900823160057651040, 6.80125930225774404579034064904, 8.083920722354665600101333824082
