| L(s) = 1 | + 10·7-s + 14·11-s − 82·13-s − 18·17-s − 136·19-s + 140·23-s − 112·29-s + 72·31-s + 26·37-s + 446·41-s + 396·43-s + 144·47-s − 243·49-s − 158·53-s + 342·59-s + 314·61-s − 152·67-s + 932·71-s − 548·73-s + 140·77-s − 512·79-s − 284·83-s + 810·89-s − 820·91-s + 1.30e3·97-s − 936·101-s + 1.45e3·103-s + ⋯ |
| L(s) = 1 | + 0.539·7-s + 0.383·11-s − 1.74·13-s − 0.256·17-s − 1.64·19-s + 1.26·23-s − 0.717·29-s + 0.417·31-s + 0.115·37-s + 1.69·41-s + 1.40·43-s + 0.446·47-s − 0.708·49-s − 0.409·53-s + 0.754·59-s + 0.659·61-s − 0.277·67-s + 1.55·71-s − 0.878·73-s + 0.207·77-s − 0.729·79-s − 0.375·83-s + 0.964·89-s − 0.944·91-s + 1.36·97-s − 0.922·101-s + 1.38·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.854209948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.854209948\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−9.014859810853949208531482808218, −8.080137290316918021756499849738, −7.33879740347375190184117061773, −6.64101098032686018223809665583, −5.63020478749622880711644906147, −4.71547665426099448799340737053, −4.14356032412627182310832613743, −2.73972206346481273554822818238, −2.00171616250928604973915910155, −0.62605754925260666432454304724,
0.62605754925260666432454304724, 2.00171616250928604973915910155, 2.73972206346481273554822818238, 4.14356032412627182310832613743, 4.71547665426099448799340737053, 5.63020478749622880711644906147, 6.64101098032686018223809665583, 7.33879740347375190184117061773, 8.080137290316918021756499849738, 9.014859810853949208531482808218
