| L(s) = 1 | − 9·9-s + 72·11-s + 200·19-s + 468·29-s − 32·31-s + 180·41-s + 622·49-s + 1.36e3·59-s + 844·61-s − 720·71-s − 1.02e3·79-s + 81·81-s + 1.26e3·89-s − 648·99-s + 1.11e3·101-s − 3.24e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.29e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.97·11-s + 2.41·19-s + 2.99·29-s − 0.185·31-s + 0.685·41-s + 1.81·49-s + 3.01·59-s + 1.77·61-s − 1.20·71-s − 1.45·79-s + 1/9·81-s + 1.50·89-s − 0.657·99-s + 1.09·101-s − 2.85·109-s + 0.921·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.95·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.817290148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.817290148\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 622 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4294 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9502 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 50230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 21022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 684 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 491302 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 360 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 777358 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 512 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 267770 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 714430 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.51536405683585010970300149040, −11.46034422609043977428802931929, −10.30149725236378895947524459441, −10.30123159622359384634317176678, −9.614038276757062800617854573237, −9.201726319708660455400379578182, −8.658083341052038512888120794330, −8.440588393978144737884839762398, −7.51242743499894942596596104057, −7.25918426715692784224466813855, −6.45189725022693994664513603490, −6.40216532635018903090561939321, −5.33101338037552843692236606174, −5.23191986077072422926754362984, −4.13521002902503709649179080873, −3.89777376088361802418684514920, −3.03226289994440797880429787985, −2.45927099866242398519438737243, −1.07799533821304888474919268607, −0.998695260928574137132394058307,
0.998695260928574137132394058307, 1.07799533821304888474919268607, 2.45927099866242398519438737243, 3.03226289994440797880429787985, 3.89777376088361802418684514920, 4.13521002902503709649179080873, 5.23191986077072422926754362984, 5.33101338037552843692236606174, 6.40216532635018903090561939321, 6.45189725022693994664513603490, 7.25918426715692784224466813855, 7.51242743499894942596596104057, 8.440588393978144737884839762398, 8.658083341052038512888120794330, 9.201726319708660455400379578182, 9.614038276757062800617854573237, 10.30123159622359384634317176678, 10.30149725236378895947524459441, 11.46034422609043977428802931929, 11.51536405683585010970300149040
