| L(s) = 1 | + 3·3-s + 5·5-s − 7·7-s + 9·9-s − 44·11-s + 22·13-s + 15·15-s + 66·17-s − 132·19-s − 21·21-s − 168·23-s + 25·25-s + 27·27-s + 54·29-s + 144·31-s − 132·33-s − 35·35-s − 354·37-s + 66·39-s − 22·41-s − 156·43-s + 45·45-s − 240·47-s + 49·49-s + 198·51-s − 354·53-s − 220·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.469·13-s + 0.258·15-s + 0.941·17-s − 1.59·19-s − 0.218·21-s − 1.52·23-s + 1/5·25-s + 0.192·27-s + 0.345·29-s + 0.834·31-s − 0.696·33-s − 0.169·35-s − 1.57·37-s + 0.270·39-s − 0.0838·41-s − 0.553·43-s + 0.149·45-s − 0.744·47-s + 1/7·49-s + 0.543·51-s − 0.917·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 132 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 - 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 354 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 156 T + p^{3} T^{2} \) |
| 47 | \( 1 + 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 + 76 T + p^{3} T^{2} \) |
| 61 | \( 1 + 154 T + p^{3} T^{2} \) |
| 67 | \( 1 + 628 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1018 T + p^{3} T^{2} \) |
| 79 | \( 1 - 96 T + p^{3} T^{2} \) |
| 83 | \( 1 - 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 218 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1598 T + p^{3} T^{2} \) |
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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
−9.463112204626031341325014868354, −8.352223059285472737363868070695, −7.990451343038805971525958490289, −6.72663066865084324026620794103, −5.95208239355464202701394914454, −4.90732816925327008681566393021, −3.74483114097115153235936848994, −2.73242483884150337773736941940, −1.72300840079941920792412613461, 0,
1.72300840079941920792412613461, 2.73242483884150337773736941940, 3.74483114097115153235936848994, 4.90732816925327008681566393021, 5.95208239355464202701394914454, 6.72663066865084324026620794103, 7.990451343038805971525958490289, 8.352223059285472737363868070695, 9.463112204626031341325014868354
