| L(s) = 1 | + 2·3-s − 9-s − 6·13-s − 24·17-s + 16·23-s + 7·25-s − 2·27-s − 2·29-s − 12·39-s + 22·43-s − 2·49-s − 48·51-s − 22·53-s − 2·61-s + 32·69-s + 14·75-s + 52·79-s − 7·81-s − 4·87-s + 50·101-s − 28·103-s + 4·107-s + 20·113-s + 6·117-s + 31·121-s + 127-s + 44·129-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s − 1.66·13-s − 5.82·17-s + 3.33·23-s + 7/5·25-s − 0.384·27-s − 0.371·29-s − 1.92·39-s + 3.35·43-s − 2/7·49-s − 6.72·51-s − 3.02·53-s − 0.256·61-s + 3.85·69-s + 1.61·75-s + 5.85·79-s − 7/9·81-s − 0.428·87-s + 4.97·101-s − 2.75·103-s + 0.386·107-s + 1.88·113-s + 0.554·117-s + 2.81·121-s + 0.0887·127-s + 3.87·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.508798148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.508798148\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 31 T^{2} + 444 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 - 67 T^{2} + 1840 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 45 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 58 T^{2} + 1675 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 31 T^{2} - 120 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 63 T^{2} + 2480 T^{4} - 63 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 69 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 11 T + 132 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 14814 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 115 T^{2} + 11056 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 9614 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 58 T^{2} + 4699 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 231 T^{2} + 25244 T^{4} - 231 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 263 T^{2} + 32416 T^{4} - 263 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.33255965249625910999521434639, −6.27641652447238219893639443324, −5.80305867402726141102261925017, −5.59902544041957103347666697102, −5.20268295724016456013212898575, −5.04852322380671087981253813453, −4.85409018089960291775571923328, −4.68361665386937224647329286544, −4.63597715661717503160105596267, −4.51279931501406285599175031791, −4.19253026979241543324608843808, −3.91705494668552329432139043072, −3.73793241081285085140808925650, −3.19466327631324011828943090070, −3.02368977222260878516434512530, −2.97608803665679315336573721478, −2.73785820156253385616283264722, −2.60960394761206804581410578668, −2.18544346230986723050374065561, −2.00699120402643131390918923716, −1.94774048632710072580048180053, −1.68064732704385411269065571295, −0.70795953675900111646003694648, −0.56824377115810472568419321296, −0.55376665105568407338622809848,
0.55376665105568407338622809848, 0.56824377115810472568419321296, 0.70795953675900111646003694648, 1.68064732704385411269065571295, 1.94774048632710072580048180053, 2.00699120402643131390918923716, 2.18544346230986723050374065561, 2.60960394761206804581410578668, 2.73785820156253385616283264722, 2.97608803665679315336573721478, 3.02368977222260878516434512530, 3.19466327631324011828943090070, 3.73793241081285085140808925650, 3.91705494668552329432139043072, 4.19253026979241543324608843808, 4.51279931501406285599175031791, 4.63597715661717503160105596267, 4.68361665386937224647329286544, 4.85409018089960291775571923328, 5.04852322380671087981253813453, 5.20268295724016456013212898575, 5.59902544041957103347666697102, 5.80305867402726141102261925017, 6.27641652447238219893639443324, 6.33255965249625910999521434639
