| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s − 6·11-s − 2·13-s + 2·15-s + 19-s − 21-s − 4·23-s − 25-s + 27-s − 6·29-s − 2·31-s − 6·33-s − 2·35-s + 8·37-s − 2·39-s + 2·41-s − 12·43-s + 2·45-s − 2·47-s + 49-s − 6·53-s − 12·55-s + 57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.516·15-s + 0.229·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.359·31-s − 1.04·33-s − 0.338·35-s + 1.31·37-s − 0.320·39-s + 0.312·41-s − 1.82·43-s + 0.298·45-s − 0.291·47-s + 1/7·49-s − 0.824·53-s − 1.61·55-s + 0.132·57-s + 0.520·59-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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.132770459357843418164990123156, −7.73273276558545259656910693703, −6.87451850585610869684789000449, −5.87207074864525699534951197047, −5.36848379387023208489438805501, −4.43947704735137258758094408778, −3.29045433126333024879007161403, −2.52322160109349990708835269837, −1.80889703169017167144916668232, 0,
1.80889703169017167144916668232, 2.52322160109349990708835269837, 3.29045433126333024879007161403, 4.43947704735137258758094408778, 5.36848379387023208489438805501, 5.87207074864525699534951197047, 6.87451850585610869684789000449, 7.73273276558545259656910693703, 8.132770459357843418164990123156
