| L(s) = 1 | + 2·5-s − 2·11-s − 4·13-s + 19-s − 8·23-s − 25-s − 2·29-s + 2·31-s − 8·37-s − 2·41-s − 4·43-s + 4·47-s − 7·49-s + 2·53-s − 4·55-s − 10·61-s − 8·65-s + 16·71-s + 6·73-s − 14·79-s + 6·83-s − 18·89-s + 2·95-s + 10·97-s + 14·101-s − 6·103-s − 8·107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.603·11-s − 1.10·13-s + 0.229·19-s − 1.66·23-s − 1/5·25-s − 0.371·29-s + 0.359·31-s − 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 49-s + 0.274·53-s − 0.539·55-s − 1.28·61-s − 0.992·65-s + 1.89·71-s + 0.702·73-s − 1.57·79-s + 0.658·83-s − 1.90·89-s + 0.205·95-s + 1.01·97-s + 1.39·101-s − 0.591·103-s − 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.391198236506038957221629773697, −7.72097144375492209353190700554, −6.92076782170002683072135481731, −6.04537670491531984013901490450, −5.39006577777807946122287060730, −4.65486938055172924173421055502, −3.53711099784593745581971184422, −2.45080129749770972997412968234, −1.75738623548621053409146805722, 0,
1.75738623548621053409146805722, 2.45080129749770972997412968234, 3.53711099784593745581971184422, 4.65486938055172924173421055502, 5.39006577777807946122287060730, 6.04537670491531984013901490450, 6.92076782170002683072135481731, 7.72097144375492209353190700554, 8.391198236506038957221629773697
