| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 3·7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 3·14-s + 15-s + 16-s − 18-s + 5·19-s − 20-s − 3·21-s + 22-s − 4·23-s + 24-s + 25-s − 27-s + 3·28-s − 30-s − 10·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.566·28-s − 0.182·30-s − 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.83964863322762188092883445743, −7.38296364446977614928003210959, −6.59281974936670294299730115426, −5.58083830356357407411598168677, −5.13202677683347561333967431902, −4.18805425242667360026731729046, −3.28268674688962100617480697604, −2.05874485144909035505789823909, −1.24907580045746988515620789018, 0,
1.24907580045746988515620789018, 2.05874485144909035505789823909, 3.28268674688962100617480697604, 4.18805425242667360026731729046, 5.13202677683347561333967431902, 5.58083830356357407411598168677, 6.59281974936670294299730115426, 7.38296364446977614928003210959, 7.83964863322762188092883445743
