| L(s) = 1 | + 3·3-s + 10·7-s + 9·9-s − 14·11-s − 82·13-s + 18·17-s − 136·19-s + 30·21-s − 140·23-s + 27·27-s + 112·29-s + 72·31-s − 42·33-s + 26·37-s − 246·39-s − 446·41-s + 396·43-s − 144·47-s − 243·49-s + 54·51-s + 158·53-s − 408·57-s − 342·59-s + 314·61-s + 90·63-s − 152·67-s − 420·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.539·7-s + 1/3·9-s − 0.383·11-s − 1.74·13-s + 0.256·17-s − 1.64·19-s + 0.311·21-s − 1.26·23-s + 0.192·27-s + 0.717·29-s + 0.417·31-s − 0.221·33-s + 0.115·37-s − 1.01·39-s − 1.69·41-s + 1.40·43-s − 0.446·47-s − 0.708·49-s + 0.148·51-s + 0.409·53-s − 0.948·57-s − 0.754·59-s + 0.659·61-s + 0.179·63-s − 0.277·67-s − 0.732·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 + 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 136 T + p^{3} T^{2} \) |
| 23 | \( 1 + 140 T + p^{3} T^{2} \) |
| 29 | \( 1 - 112 T + p^{3} T^{2} \) |
| 31 | \( 1 - 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 446 T + p^{3} T^{2} \) |
| 43 | \( 1 - 396 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 158 T + p^{3} T^{2} \) |
| 59 | \( 1 + 342 T + p^{3} T^{2} \) |
| 61 | \( 1 - 314 T + p^{3} T^{2} \) |
| 67 | \( 1 + 152 T + p^{3} T^{2} \) |
| 71 | \( 1 + 932 T + p^{3} T^{2} \) |
| 73 | \( 1 + 548 T + p^{3} T^{2} \) |
| 79 | \( 1 + 512 T + p^{3} T^{2} \) |
| 83 | \( 1 - 284 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1304 T + p^{3} T^{2} \) |
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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
−9.996212525013104910012155093589, −8.827560756351837793419150428842, −8.065542766505303470110054412559, −7.34921525731902082441982806981, −6.25133811720376164162224463118, −4.97316070405265128123452208969, −4.22429663654816161189254778080, −2.76333385498009938861823142694, −1.88396817716477764064835813873, 0,
1.88396817716477764064835813873, 2.76333385498009938861823142694, 4.22429663654816161189254778080, 4.97316070405265128123452208969, 6.25133811720376164162224463118, 7.34921525731902082441982806981, 8.065542766505303470110054412559, 8.827560756351837793419150428842, 9.996212525013104910012155093589
