| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s + 8-s + 1.34·9-s − 1.87·11-s + 1.53·12-s + 16-s − 17-s + 1.34·18-s − 1.87·22-s + 1.53·24-s + 25-s + 0.532·27-s + 32-s − 2.87·33-s − 34-s + 1.34·36-s + 0.347·41-s − 43-s − 1.87·44-s + 1.53·48-s + 49-s + 50-s − 1.53·51-s + 0.532·54-s + ⋯ |
| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s + 8-s + 1.34·9-s − 1.87·11-s + 1.53·12-s + 16-s − 17-s + 1.34·18-s − 1.87·22-s + 1.53·24-s + 25-s + 0.532·27-s + 32-s − 2.87·33-s − 34-s + 1.34·36-s + 0.347·41-s − 43-s − 1.87·44-s + 1.53·48-s + 49-s + 50-s − 1.53·51-s + 0.532·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.536652047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.536652047\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.53T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.87T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 0.347T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.87T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.347T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - 0.347T + T^{2} \) |
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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
−8.735814854117361249413901260821, −8.120074861783521853868024476742, −7.48820981806210813748944614009, −6.82726721466616572431892745613, −5.74239784395384400241760929041, −4.88139765214673274116053061492, −4.18980652204260096263471002782, −3.07924806086391382239098424675, −2.70781819772934452087379043192, −1.84410440710171532302524274272,
1.84410440710171532302524274272, 2.70781819772934452087379043192, 3.07924806086391382239098424675, 4.18980652204260096263471002782, 4.88139765214673274116053061492, 5.74239784395384400241760929041, 6.82726721466616572431892745613, 7.48820981806210813748944614009, 8.120074861783521853868024476742, 8.735814854117361249413901260821
