| L(s) = 1 | + 3·3-s + 3·5-s + 2·7-s + 4·9-s − 2·11-s + 6·13-s + 9·15-s − 2·17-s + 12·19-s + 6·21-s + 2·23-s + 6·27-s + 10·29-s − 17·31-s − 6·33-s + 6·35-s + 18·39-s + 41-s + 5·43-s + 12·45-s + 20·47-s + 3·49-s − 6·51-s − 5·53-s − 6·55-s + 36·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 0.755·7-s + 4/3·9-s − 0.603·11-s + 1.66·13-s + 2.32·15-s − 0.485·17-s + 2.75·19-s + 1.30·21-s + 0.417·23-s + 1.15·27-s + 1.85·29-s − 3.05·31-s − 1.04·33-s + 1.01·35-s + 2.88·39-s + 0.156·41-s + 0.762·43-s + 1.78·45-s + 2.91·47-s + 3/7·49-s − 0.840·51-s − 0.686·53-s − 0.809·55-s + 4.76·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58003456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.64715344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.64715344\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 17 T + 131 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_4$ | \( 1 - 5 T + 89 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 109 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 125 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 151 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 221 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−7.944934034335435759817118527447, −7.910298192646679045772349001732, −7.25793677535993904653603899774, −7.19501630409875071375769717646, −6.97695603493435653851663223280, −6.13186850992989434572603760729, −5.75436756965080351393044161218, −5.74769186556250414604339508424, −5.41533131089024956351559755758, −4.83057567238028963239467971543, −4.48190328721114832074427545151, −4.02179279482565528866188796609, −3.50940808590972364699597457299, −3.29996966593222827367949656326, −2.75170727941947928227496932347, −2.66277440911666220679507396558, −1.90940138442795492615981329780, −1.80939753364485535638291314682, −1.14155459465087261794484673118, −0.839801681863907361428008732476,
0.839801681863907361428008732476, 1.14155459465087261794484673118, 1.80939753364485535638291314682, 1.90940138442795492615981329780, 2.66277440911666220679507396558, 2.75170727941947928227496932347, 3.29996966593222827367949656326, 3.50940808590972364699597457299, 4.02179279482565528866188796609, 4.48190328721114832074427545151, 4.83057567238028963239467971543, 5.41533131089024956351559755758, 5.74769186556250414604339508424, 5.75436756965080351393044161218, 6.13186850992989434572603760729, 6.97695603493435653851663223280, 7.19501630409875071375769717646, 7.25793677535993904653603899774, 7.910298192646679045772349001732, 7.944934034335435759817118527447
