| L(s) = 1 | + 1.08·3-s − 5-s − 1.08·7-s − 1.82·9-s + 2.61·11-s + 0.828·13-s − 1.08·15-s + 17-s − 5.22·19-s − 1.17·21-s − 1.08·23-s + 25-s − 5.22·27-s + 3.65·29-s − 2.61·31-s + 2.82·33-s + 1.08·35-s − 7.65·37-s + 0.896·39-s − 2·41-s + 11.9·43-s + 1.82·45-s − 11.0·47-s − 5.82·49-s + 1.08·51-s − 13.3·53-s − 2.61·55-s + ⋯ |
| L(s) = 1 | + 0.624·3-s − 0.447·5-s − 0.409·7-s − 0.609·9-s + 0.787·11-s + 0.229·13-s − 0.279·15-s + 0.242·17-s − 1.19·19-s − 0.255·21-s − 0.225·23-s + 0.200·25-s − 1.00·27-s + 0.679·29-s − 0.469·31-s + 0.492·33-s + 0.182·35-s − 1.25·37-s + 0.143·39-s − 0.312·41-s + 1.82·43-s + 0.272·45-s − 1.61·47-s − 0.832·49-s + 0.151·51-s − 1.82·53-s − 0.352·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 19 | \( 1 + 5.22T + 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 2.61T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 0.896T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 9.31T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.340861153790239842656343378075, −7.998641714694619499494933648679, −6.83051618078798573531246982031, −6.33907865925945060516067434394, −5.34927877701210760992403284530, −4.26334979815141422002948050169, −3.56024338994508861709764991965, −2.78447987108201536437773170433, −1.63464809842454886893717012326, 0,
1.63464809842454886893717012326, 2.78447987108201536437773170433, 3.56024338994508861709764991965, 4.26334979815141422002948050169, 5.34927877701210760992403284530, 6.33907865925945060516067434394, 6.83051618078798573531246982031, 7.998641714694619499494933648679, 8.340861153790239842656343378075
