| L(s) = 1 | − 1.73·3-s − 5-s + 1.99·9-s + 1.73·15-s − 1.73·23-s + 25-s − 1.73·27-s − 29-s + 41-s − 1.73·43-s − 1.99·45-s + 61-s + 1.73·67-s + 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s − 89-s − 101-s + 1.73·103-s + 1.73·107-s + 109-s + 1.73·115-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 5-s + 1.99·9-s + 1.73·15-s − 1.73·23-s + 25-s − 1.73·27-s − 29-s + 41-s − 1.73·43-s − 1.99·45-s + 61-s + 1.73·67-s + 2.99·69-s − 1.73·75-s + 0.999·81-s + 1.73·83-s + 1.73·87-s − 89-s − 101-s + 1.73·103-s + 1.73·107-s + 109-s + 1.73·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4286168740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4286168740\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.73T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - 1.73T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.73T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.468516270749958112318313506889, −7.76655971803882216531330490111, −7.06217667125121731686400338468, −6.38169813902501019082743582940, −5.68223021165370224851472751190, −4.95782050175129613019179083427, −4.21143610443135802774432277955, −3.52953135609114500770691446304, −1.95033197646336209607277874156, −0.59689074176784338880675773165,
0.59689074176784338880675773165, 1.95033197646336209607277874156, 3.52953135609114500770691446304, 4.21143610443135802774432277955, 4.95782050175129613019179083427, 5.68223021165370224851472751190, 6.38169813902501019082743582940, 7.06217667125121731686400338468, 7.76655971803882216531330490111, 8.468516270749958112318313506889
