| L(s) = 1 | − 27·9-s − 92·13-s + 104·17-s + 130·29-s + 396·37-s + 230·41-s − 343·49-s + 572·53-s − 830·61-s + 592·73-s + 729·81-s + 1.67e3·89-s + 1.81e3·97-s + 598·101-s − 1.74e3·109-s + 1.32e3·113-s + 2.48e3·117-s + ⋯ |
| L(s) = 1 | − 9-s − 1.96·13-s + 1.48·17-s + 0.832·29-s + 1.75·37-s + 0.876·41-s − 49-s + 1.48·53-s − 1.74·61-s + 0.949·73-s + 81-s + 1.98·89-s + 1.90·97-s + 0.589·101-s − 1.53·109-s + 1.10·113-s + 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.534950159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534950159\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 92 T + p^{3} T^{2} \) |
| 17 | \( 1 - 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 396 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 592 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−9.826330107220046416643218109512, −9.163695757124538940126454600345, −7.984700761173765800339336383560, −7.51096866893660098059935653888, −6.29917929288157783502469485986, −5.40999718762615467838351540179, −4.58917072611100257682175456072, −3.17887520547333551337684784309, −2.36771612971197899888958510063, −0.67342500078724379311019811573,
0.67342500078724379311019811573, 2.36771612971197899888958510063, 3.17887520547333551337684784309, 4.58917072611100257682175456072, 5.40999718762615467838351540179, 6.29917929288157783502469485986, 7.51096866893660098059935653888, 7.984700761173765800339336383560, 9.163695757124538940126454600345, 9.826330107220046416643218109512
