| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s − 2·13-s + 4·15-s + 4·17-s + 19-s + 21-s + 4·23-s + 11·25-s − 27-s + 4·29-s + 4·35-s − 10·37-s + 2·39-s + 10·41-s + 4·43-s − 4·45-s − 6·47-s + 49-s − 4·51-s − 57-s + 4·59-s − 10·61-s − 63-s + 8·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 11/5·25-s − 0.192·27-s + 0.742·29-s + 0.676·35-s − 1.64·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.596·45-s − 0.875·47-s + 1/7·49-s − 0.560·51-s − 0.132·57-s + 0.520·59-s − 1.28·61-s − 0.125·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.186808495375927710290517779319, −7.36380136998996332897085914758, −7.10245175705742542456471160146, −6.03924896184943598627006249770, −5.08084089218605196627138110393, −4.43021844461524779947464379669, −3.56418100801958708410840009819, −2.86583893479602789304544497029, −1.08895311619327121384054030083, 0,
1.08895311619327121384054030083, 2.86583893479602789304544497029, 3.56418100801958708410840009819, 4.43021844461524779947464379669, 5.08084089218605196627138110393, 6.03924896184943598627006249770, 7.10245175705742542456471160146, 7.36380136998996332897085914758, 8.186808495375927710290517779319
