| L(s) = 1 | + 3·4-s − 9-s − 2·11-s + 5·16-s + 16·19-s − 2·29-s + 6·31-s − 3·36-s − 4·41-s − 6·44-s + 5·49-s − 10·59-s + 2·61-s + 3·64-s + 32·71-s + 48·76-s − 24·79-s + 81-s + 2·99-s + 18·101-s + 20·109-s − 6·116-s − 19·121-s + 18·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s − 0.603·11-s + 5/4·16-s + 3.67·19-s − 0.371·29-s + 1.07·31-s − 1/2·36-s − 0.624·41-s − 0.904·44-s + 5/7·49-s − 1.30·59-s + 0.256·61-s + 3/8·64-s + 3.79·71-s + 5.50·76-s − 2.70·79-s + 1/9·81-s + 0.201·99-s + 1.79·101-s + 1.91·109-s − 0.557·116-s − 1.72·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.398196703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398196703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.993365486017427421181187407919, −9.953219325022572253019619379447, −9.610572909105518852360956890047, −8.931870849238938903462290077281, −8.548589806551599059976766094615, −7.81579865387732758228568223813, −7.69713105689630168693199699420, −7.34426558003431121544280184614, −6.88062157319066545505867244925, −6.48066467962313670565829759637, −5.74135727289181725149398634475, −5.70509225015289243159030107494, −5.02701032876611107425291241719, −4.72309955422586542185131622997, −3.52997842255030820555157743329, −3.44831572882204286425184494551, −2.74840290685116707800225510351, −2.42895203304094878911833113373, −1.53357364983604927373442074658, −0.902528051191669292261780888325,
0.902528051191669292261780888325, 1.53357364983604927373442074658, 2.42895203304094878911833113373, 2.74840290685116707800225510351, 3.44831572882204286425184494551, 3.52997842255030820555157743329, 4.72309955422586542185131622997, 5.02701032876611107425291241719, 5.70509225015289243159030107494, 5.74135727289181725149398634475, 6.48066467962313670565829759637, 6.88062157319066545505867244925, 7.34426558003431121544280184614, 7.69713105689630168693199699420, 7.81579865387732758228568223813, 8.548589806551599059976766094615, 8.931870849238938903462290077281, 9.610572909105518852360956890047, 9.953219325022572253019619379447, 9.993365486017427421181187407919
