| L(s) = 1 | − 3·3-s + 5·5-s − 4·7-s + 9·9-s + 44·11-s + 58·13-s − 15·15-s − 62·17-s − 19·19-s + 12·21-s − 28·23-s + 25·25-s − 27·27-s − 202·29-s − 320·31-s − 132·33-s − 20·35-s − 30·37-s − 174·39-s − 342·41-s + 328·43-s + 45·45-s + 236·47-s − 327·49-s + 186·51-s − 102·53-s + 220·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.215·7-s + 1/3·9-s + 1.20·11-s + 1.23·13-s − 0.258·15-s − 0.884·17-s − 0.229·19-s + 0.124·21-s − 0.253·23-s + 1/5·25-s − 0.192·27-s − 1.29·29-s − 1.85·31-s − 0.696·33-s − 0.0965·35-s − 0.133·37-s − 0.714·39-s − 1.30·41-s + 1.16·43-s + 0.149·45-s + 0.732·47-s − 0.953·49-s + 0.510·51-s − 0.264·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 29 | \( 1 + 202 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 342 T + p^{3} T^{2} \) |
| 43 | \( 1 - 328 T + p^{3} T^{2} \) |
| 47 | \( 1 - 236 T + p^{3} T^{2} \) |
| 53 | \( 1 + 102 T + p^{3} T^{2} \) |
| 59 | \( 1 - 268 T + p^{3} T^{2} \) |
| 61 | \( 1 - 502 T + p^{3} T^{2} \) |
| 67 | \( 1 + 556 T + p^{3} T^{2} \) |
| 71 | \( 1 - 192 T + p^{3} T^{2} \) |
| 73 | \( 1 - 546 T + p^{3} T^{2} \) |
| 79 | \( 1 - 144 T + p^{3} T^{2} \) |
| 83 | \( 1 - 304 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 890 T + p^{3} T^{2} \) |
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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
−8.463723764639760315110546911370, −7.27767183887403181729294440704, −6.59880351656048092817662961250, −5.99932041575393538165200770932, −5.27579755520100394801301978842, −4.09007578925885217090297995373, −3.58725277634475631284409503845, −2.06720531961076844810614394111, −1.28064428617827954902932406442, 0,
1.28064428617827954902932406442, 2.06720531961076844810614394111, 3.58725277634475631284409503845, 4.09007578925885217090297995373, 5.27579755520100394801301978842, 5.99932041575393538165200770932, 6.59880351656048092817662961250, 7.27767183887403181729294440704, 8.463723764639760315110546911370
