| L(s) = 1 | − 3·5-s − 2·7-s − 5·11-s − 2·13-s + 5·17-s + 9·19-s + 3·23-s + 5·25-s − 15·29-s − 10·31-s + 6·35-s + 5·37-s + 12·41-s + 9·43-s + 4·47-s + 3·49-s − 6·53-s + 15·55-s − 6·59-s − 7·61-s + 6·65-s + 24·67-s − 20·71-s + 13·73-s + 10·77-s − 18·79-s + 14·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.755·7-s − 1.50·11-s − 0.554·13-s + 1.21·17-s + 2.06·19-s + 0.625·23-s + 25-s − 2.78·29-s − 1.79·31-s + 1.01·35-s + 0.821·37-s + 1.87·41-s + 1.37·43-s + 0.583·47-s + 3/7·49-s − 0.824·53-s + 2.02·55-s − 0.781·59-s − 0.896·61-s + 0.744·65-s + 2.93·67-s − 2.37·71-s + 1.52·73-s + 1.13·77-s − 2.02·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42928704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42928704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 15 T + 106 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 98 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 180 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 206 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 182 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.55427909697988221866632301963, −7.51697618624304333324865427926, −7.26505107913210410167804153663, −7.24652779700113212882652137575, −6.39936279073400702802731750378, −5.88625418611655523484049834569, −5.63154240607427381068309079110, −5.40459195550528532357407695594, −5.06285038021067423666291664724, −4.63772701242220749168725415090, −3.89386391665239484029229981933, −3.88080275562419193051011418749, −3.40352900719775832249539485296, −3.08411698014376635450029956947, −2.50859500788490083314487380344, −2.40161638388589515391796808353, −1.33994444912205386639549027614, −1.00052919363521109336459018594, 0, 0,
1.00052919363521109336459018594, 1.33994444912205386639549027614, 2.40161638388589515391796808353, 2.50859500788490083314487380344, 3.08411698014376635450029956947, 3.40352900719775832249539485296, 3.88080275562419193051011418749, 3.89386391665239484029229981933, 4.63772701242220749168725415090, 5.06285038021067423666291664724, 5.40459195550528532357407695594, 5.63154240607427381068309079110, 5.88625418611655523484049834569, 6.39936279073400702802731750378, 7.24652779700113212882652137575, 7.26505107913210410167804153663, 7.51697618624304333324865427926, 7.55427909697988221866632301963
