| L(s) = 1 | − 9-s + 12·11-s + 2·19-s + 12·29-s + 6·31-s + 8·41-s − 11·49-s − 12·59-s − 6·61-s − 24·71-s + 16·79-s + 81-s + 32·89-s − 12·99-s + 16·101-s − 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 3.61·11-s + 0.458·19-s + 2.22·29-s + 1.07·31-s + 1.24·41-s − 1.57·49-s − 1.56·59-s − 0.768·61-s − 2.84·71-s + 1.80·79-s + 1/9·81-s + 3.39·89-s − 1.20·99-s + 1.59·101-s − 1.34·109-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.543039956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.543039956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−8.584753092487217925910789315590, −8.070386165258264988331779367625, −7.73568343271214195814678494758, −7.45444409110125013809191357255, −6.73642010792931587839690470639, −6.65214468058777236027927195718, −6.26422861346898765050741812747, −6.18874262511404438246195101406, −5.76164771756570215256683818429, −4.89018878615758594764240389838, −4.73140944595475011944813448837, −4.43643856529949201010740593328, −3.90278830921214390507713179810, −3.62655754323216977496943798067, −3.10268610706440176264105835414, −2.81703981853751883824249755605, −2.07188570304636866346138555998, −1.42464396943373685245798694693, −1.21221986866717236464162381646, −0.66526899198926756244062975514,
0.66526899198926756244062975514, 1.21221986866717236464162381646, 1.42464396943373685245798694693, 2.07188570304636866346138555998, 2.81703981853751883824249755605, 3.10268610706440176264105835414, 3.62655754323216977496943798067, 3.90278830921214390507713179810, 4.43643856529949201010740593328, 4.73140944595475011944813448837, 4.89018878615758594764240389838, 5.76164771756570215256683818429, 6.18874262511404438246195101406, 6.26422861346898765050741812747, 6.65214468058777236027927195718, 6.73642010792931587839690470639, 7.45444409110125013809191357255, 7.73568343271214195814678494758, 8.070386165258264988331779367625, 8.584753092487217925910789315590
