| L(s) = 1 | + 3-s + 2·5-s + 5·7-s − 9-s + 8·11-s − 13-s + 2·15-s + 3·17-s + 2·19-s + 5·21-s + 23-s + 3·25-s + 11·29-s − 2·31-s + 8·33-s + 10·35-s − 6·37-s − 39-s + 4·43-s − 2·45-s − 2·47-s + 9·49-s + 3·51-s − 9·53-s + 16·55-s + 2·57-s − 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.88·7-s − 1/3·9-s + 2.41·11-s − 0.277·13-s + 0.516·15-s + 0.727·17-s + 0.458·19-s + 1.09·21-s + 0.208·23-s + 3/5·25-s + 2.04·29-s − 0.359·31-s + 1.39·33-s + 1.69·35-s − 0.986·37-s − 0.160·39-s + 0.609·43-s − 0.298·45-s − 0.291·47-s + 9/7·49-s + 0.420·51-s − 1.23·53-s + 2.15·55-s + 0.264·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.584176800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.584176800\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 84 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 122 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 186 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 17 T + 202 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 60 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.315507256576916796396892464541, −7.969651075274051660305262672216, −7.73481058444903123996088386283, −6.94535523318775817339876394312, −6.86122411739726078262032028020, −6.70848009831705645681524931654, −5.93249128047939885354153600072, −5.92937235678922846082190471563, −5.23427759648062209665367516142, −5.02318358195042063056795851237, −4.70884087838282243015241458265, −4.29270344077333594334145137561, −3.69618932089459072350032122476, −3.59370650440738886990955010702, −2.88509775242745812307144029991, −2.54399362690727469357695014806, −1.98432254075896860655833417497, −1.53306908255023097746391943587, −1.24543955199195850448238606316, −0.832187340095773646565405038223,
0.832187340095773646565405038223, 1.24543955199195850448238606316, 1.53306908255023097746391943587, 1.98432254075896860655833417497, 2.54399362690727469357695014806, 2.88509775242745812307144029991, 3.59370650440738886990955010702, 3.69618932089459072350032122476, 4.29270344077333594334145137561, 4.70884087838282243015241458265, 5.02318358195042063056795851237, 5.23427759648062209665367516142, 5.92937235678922846082190471563, 5.93249128047939885354153600072, 6.70848009831705645681524931654, 6.86122411739726078262032028020, 6.94535523318775817339876394312, 7.73481058444903123996088386283, 7.969651075274051660305262672216, 8.315507256576916796396892464541
