| L(s) = 1 | + 2-s − 3·3-s − 7·4-s − 3·6-s + 2·7-s − 15·8-s + 9·9-s − 48·11-s + 21·12-s + 14·13-s + 2·14-s + 41·16-s + 17·17-s + 9·18-s + 92·19-s − 6·21-s − 48·22-s + 122·23-s + 45·24-s + 14·26-s − 27·27-s − 14·28-s − 36·29-s − 182·31-s + 161·32-s + 144·33-s + 17·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.204·6-s + 0.107·7-s − 0.662·8-s + 1/3·9-s − 1.31·11-s + 0.505·12-s + 0.298·13-s + 0.0381·14-s + 0.640·16-s + 0.242·17-s + 0.117·18-s + 1.11·19-s − 0.0623·21-s − 0.465·22-s + 1.10·23-s + 0.382·24-s + 0.105·26-s − 0.192·27-s − 0.0944·28-s − 0.230·29-s − 1.05·31-s + 0.889·32-s + 0.759·33-s + 0.0857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 122 T + p^{3} T^{2} \) |
| 29 | \( 1 + 36 T + p^{3} T^{2} \) |
| 31 | \( 1 + 182 T + p^{3} T^{2} \) |
| 37 | \( 1 + 76 T + p^{3} T^{2} \) |
| 41 | \( 1 - 294 T + p^{3} T^{2} \) |
| 43 | \( 1 - 428 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 - 234 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 820 T + p^{3} T^{2} \) |
| 67 | \( 1 + 700 T + p^{3} T^{2} \) |
| 71 | \( 1 - 794 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 79 | \( 1 - 858 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 710 T + p^{3} T^{2} \) |
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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
−9.085947962424957233228468254751, −7.945037163559729410882469179557, −7.38318166632617314076489936736, −6.08904897795350236406836497966, −5.34456563781486096313207021227, −4.85732439664041751696266099206, −3.74453756399761346483875763335, −2.78104058504585025893359721349, −1.12082469356584300253359302357, 0,
1.12082469356584300253359302357, 2.78104058504585025893359721349, 3.74453756399761346483875763335, 4.85732439664041751696266099206, 5.34456563781486096313207021227, 6.08904897795350236406836497966, 7.38318166632617314076489936736, 7.945037163559729410882469179557, 9.085947962424957233228468254751
