| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 5·11-s + 5·13-s − 15-s − 5·17-s − 21-s − 8·23-s + 25-s + 5·27-s − 29-s − 2·31-s + 5·33-s + 35-s + 4·37-s − 5·39-s + 2·41-s − 4·43-s − 2·45-s − 13·47-s + 49-s + 5·51-s − 8·53-s − 5·55-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 1.38·13-s − 0.258·15-s − 1.21·17-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s − 0.359·31-s + 0.870·33-s + 0.169·35-s + 0.657·37-s − 0.800·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.89·47-s + 1/7·49-s + 0.700·51-s − 1.09·53-s − 0.674·55-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−9.491948335238814850919165746875, −8.370051000148633364855371195177, −8.063517141148345392043405722751, −6.66748854828879212944424240318, −5.94481578275946266777055244953, −5.30049955232884599582486687984, −4.29722534435929022081812306925, −2.95752036175622615716558765064, −1.82911173692261649750792126223, 0,
1.82911173692261649750792126223, 2.95752036175622615716558765064, 4.29722534435929022081812306925, 5.30049955232884599582486687984, 5.94481578275946266777055244953, 6.66748854828879212944424240318, 8.063517141148345392043405722751, 8.370051000148633364855371195177, 9.491948335238814850919165746875
