| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 2·11-s − 13-s + 15-s − 17-s + 2·19-s + 4·21-s + 25-s − 27-s − 2·29-s + 4·31-s + 2·33-s + 4·35-s − 12·37-s + 39-s − 2·41-s + 12·43-s − 45-s + 12·47-s + 9·49-s + 51-s + 2·53-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.348·33-s + 0.676·35-s − 1.97·37-s + 0.160·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 1.75·47-s + 9/7·49-s + 0.140·51-s + 0.274·53-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6007899834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6007899834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.25933286090086, −13.97051842490726, −13.30813662023559, −12.83069071186207, −12.36344831735868, −12.09320153307624, −11.38616049878674, −10.78456648092379, −10.37902768483622, −9.858587760275930, −9.360519637825770, −8.793240760925442, −8.179480737044367, −7.295127904260989, −7.189515023235191, −6.488717875605667, −5.923752485080360, −5.380015395087458, −4.828340211858520, −3.936014806135213, −3.629557709082182, −2.777667243940228, −2.307741619356156, −1.098958194917366, −0.3154503502367808,
0.3154503502367808, 1.098958194917366, 2.307741619356156, 2.777667243940228, 3.629557709082182, 3.936014806135213, 4.828340211858520, 5.380015395087458, 5.923752485080360, 6.488717875605667, 7.189515023235191, 7.295127904260989, 8.179480737044367, 8.793240760925442, 9.360519637825770, 9.858587760275930, 10.37902768483622, 10.78456648092379, 11.38616049878674, 12.09320153307624, 12.36344831735868, 12.83069071186207, 13.30813662023559, 13.97051842490726, 14.25933286090086
