| L(s) = 1 | + 5-s − 2·7-s + 2·11-s − 9·13-s − 2·19-s + 5·23-s − 5·25-s + 15·29-s − 31-s − 2·35-s + 3·37-s − 19·41-s + 2·43-s − 3·47-s + 3·49-s − 53-s + 2·55-s − 8·59-s − 9·65-s + 19·67-s − 71-s − 6·73-s − 4·77-s + 13·79-s + 83-s − 15·89-s + 18·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.603·11-s − 2.49·13-s − 0.458·19-s + 1.04·23-s − 25-s + 2.78·29-s − 0.179·31-s − 0.338·35-s + 0.493·37-s − 2.96·41-s + 0.304·43-s − 0.437·47-s + 3/7·49-s − 0.137·53-s + 0.269·55-s − 1.04·59-s − 1.11·65-s + 2.32·67-s − 0.118·71-s − 0.702·73-s − 0.455·77-s + 1.46·79-s + 0.109·83-s − 1.58·89-s + 1.88·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.196958552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196958552\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9 T + 42 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 110 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 72 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 19 T + 168 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 19 T + 186 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 138 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 13 T + 196 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 226 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.185037566124457454187965356609, −8.116449456765153032038800563412, −7.28335805555971787475607674676, −7.15250902348865712865562849079, −6.86179672765516655668586464395, −6.60745435867943226993054356340, −6.07471152038805856006886508985, −5.95884525200823944220628810270, −5.16660773076303965196930016671, −5.06619148999116622694740139628, −4.65241313871062041872323115592, −4.46123939242294234878767897192, −3.73985436468492827761787321342, −3.39174930978679927647151231476, −2.91645301135024965335950802141, −2.58925734690730868482804582221, −2.17154085802695561029573545373, −1.69378607093471906529247316726, −0.995088022773706210693827393999, −0.30485207844741623894514425027,
0.30485207844741623894514425027, 0.995088022773706210693827393999, 1.69378607093471906529247316726, 2.17154085802695561029573545373, 2.58925734690730868482804582221, 2.91645301135024965335950802141, 3.39174930978679927647151231476, 3.73985436468492827761787321342, 4.46123939242294234878767897192, 4.65241313871062041872323115592, 5.06619148999116622694740139628, 5.16660773076303965196930016671, 5.95884525200823944220628810270, 6.07471152038805856006886508985, 6.60745435867943226993054356340, 6.86179672765516655668586464395, 7.15250902348865712865562849079, 7.28335805555971787475607674676, 8.116449456765153032038800563412, 8.185037566124457454187965356609
