| L(s) = 1 | − 3·3-s − 5·5-s − 16·7-s + 9·9-s + 30·11-s − 28·13-s + 15·15-s + 78·17-s − 52·19-s + 48·21-s − 23·23-s + 25·25-s − 27·27-s − 252·29-s + 200·31-s − 90·33-s + 80·35-s + 146·37-s + 84·39-s + 438·41-s − 46·43-s − 45·45-s + 588·47-s − 87·49-s − 234·51-s + 438·53-s − 150·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.863·7-s + 1/3·9-s + 0.822·11-s − 0.597·13-s + 0.258·15-s + 1.11·17-s − 0.627·19-s + 0.498·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.61·29-s + 1.15·31-s − 0.474·33-s + 0.386·35-s + 0.648·37-s + 0.344·39-s + 1.66·41-s − 0.163·43-s − 0.149·45-s + 1.82·47-s − 0.253·49-s − 0.642·51-s + 1.13·53-s − 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 23 | \( 1 + p T \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 252 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 438 T + p^{3} T^{2} \) |
| 43 | \( 1 + 46 T + p^{3} T^{2} \) |
| 47 | \( 1 - 588 T + p^{3} T^{2} \) |
| 53 | \( 1 - 438 T + p^{3} T^{2} \) |
| 59 | \( 1 + 168 T + p^{3} T^{2} \) |
| 61 | \( 1 + 586 T + p^{3} T^{2} \) |
| 67 | \( 1 + 94 T + p^{3} T^{2} \) |
| 71 | \( 1 + 30 T + p^{3} T^{2} \) |
| 73 | \( 1 + 466 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 708 T + p^{3} T^{2} \) |
| 89 | \( 1 + 600 T + p^{3} T^{2} \) |
| 97 | \( 1 + 340 T + p^{3} T^{2} \) |
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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
−8.988931668336909206423456928560, −7.79796963929166775167712527688, −7.18880438463504270389414091625, −6.24582355900323539052549603804, −5.65614452422800200950645888533, −4.42763380291387292202710981807, −3.73609323817108850876958839290, −2.60054634472017057485837556498, −1.11244333180773083472165085438, 0,
1.11244333180773083472165085438, 2.60054634472017057485837556498, 3.73609323817108850876958839290, 4.42763380291387292202710981807, 5.65614452422800200950645888533, 6.24582355900323539052549603804, 7.18880438463504270389414091625, 7.79796963929166775167712527688, 8.988931668336909206423456928560
