L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 3·9-s + 8·11-s + 4·13-s − 4·15-s + 2·17-s + 4·19-s + 4·21-s + 2·23-s + 3·25-s + 4·27-s − 10·29-s + 14·31-s + 16·33-s − 4·35-s + 6·37-s + 8·39-s − 18·41-s − 12·43-s − 6·45-s + 20·47-s − 3·49-s + 4·51-s + 6·53-s − 16·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 2.41·11-s + 1.10·13-s − 1.03·15-s + 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s + 2.51·31-s + 2.78·33-s − 0.676·35-s + 0.986·37-s + 1.28·39-s − 2.81·41-s − 1.82·43-s − 0.894·45-s + 2.91·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s − 2.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.647805143\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.647805143\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 18 T + 161 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 20 T + 192 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 97 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 136 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 164 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 165 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T - 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−8.364869358971764432486544232724, −8.186838891654689193585792915699, −7.61038407133851102709161456071, −7.33564887958641428986346291676, −6.89762682289004846920100763336, −6.82265157402010568278774762325, −6.19398484201687217814882751083, −6.01062516328945728729552727363, −5.21620180852674665440509122172, −5.15100354273576273189962222241, −4.32497321234493354863663967334, −4.25640189290561918587938977529, −3.79033189161617006723110801665, −3.63035693658830653858738988437, −3.00821290859783217813905802207, −2.90925675270995681314017191979, −1.90587536437901508729160348106, −1.64256685714035588528212738351, −1.13493705019001449289508767985, −0.78400020226473795810350419120,
0.78400020226473795810350419120, 1.13493705019001449289508767985, 1.64256685714035588528212738351, 1.90587536437901508729160348106, 2.90925675270995681314017191979, 3.00821290859783217813905802207, 3.63035693658830653858738988437, 3.79033189161617006723110801665, 4.25640189290561918587938977529, 4.32497321234493354863663967334, 5.15100354273576273189962222241, 5.21620180852674665440509122172, 6.01062516328945728729552727363, 6.19398484201687217814882751083, 6.82265157402010568278774762325, 6.89762682289004846920100763336, 7.33564887958641428986346291676, 7.61038407133851102709161456071, 8.186838891654689193585792915699, 8.364869358971764432486544232724