| L(s) = 1 | − 0.568·3-s + 5-s − 4.73·7-s − 2.67·9-s − 0.360·11-s − 5.26·13-s − 0.568·15-s + 0.370·17-s − 4.60·19-s + 2.69·21-s + 23-s + 25-s + 3.22·27-s − 0.939·29-s − 9.66·31-s + 0.204·33-s − 4.73·35-s − 3.26·37-s + 2.99·39-s + 5.29·41-s − 2.67·45-s + 1.25·47-s + 15.4·49-s − 0.210·51-s − 10.9·53-s − 0.360·55-s + 2.61·57-s + ⋯ |
| L(s) = 1 | − 0.328·3-s + 0.447·5-s − 1.79·7-s − 0.892·9-s − 0.108·11-s − 1.45·13-s − 0.146·15-s + 0.0899·17-s − 1.05·19-s + 0.587·21-s + 0.208·23-s + 0.200·25-s + 0.620·27-s − 0.174·29-s − 1.73·31-s + 0.0356·33-s − 0.800·35-s − 0.537·37-s + 0.478·39-s + 0.827·41-s − 0.399·45-s + 0.183·47-s + 2.20·49-s − 0.0295·51-s − 1.50·53-s − 0.0486·55-s + 0.346·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2737742212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2737742212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.568T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 0.360T + 11T^{2} \) |
| 13 | \( 1 + 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.370T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 29 | \( 1 + 0.939T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 + 3.26T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.745T + 73T^{2} \) |
| 79 | \( 1 + 0.415T + 79T^{2} \) |
| 83 | \( 1 + 9.26T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.70143323089370779133628438205, −7.15523083696545591076836583997, −6.31909616653627050832019757085, −5.98744510725291979363047884416, −5.22503393969518239029376545714, −4.40055633109361669453819120308, −3.30953942559812882554730294647, −2.82601092619262090789171666479, −1.94941230275309149340780591959, −0.24607605080179303119698770101,
0.24607605080179303119698770101, 1.94941230275309149340780591959, 2.82601092619262090789171666479, 3.30953942559812882554730294647, 4.40055633109361669453819120308, 5.22503393969518239029376545714, 5.98744510725291979363047884416, 6.31909616653627050832019757085, 7.15523083696545591076836583997, 7.70143323089370779133628438205
