| L(s) = 1 | + 10·7-s − 5·13-s + 8·19-s + 5·25-s − 14·31-s + 22·37-s + 13·43-s + 61·49-s + 61-s − 11·67-s − 17·73-s − 17·79-s − 50·91-s − 14·97-s − 14·103-s − 2·109-s − 22·121-s + 127-s + 131-s + 80·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 1.38·13-s + 1.83·19-s + 25-s − 2.51·31-s + 3.61·37-s + 1.98·43-s + 61/7·49-s + 0.128·61-s − 1.34·67-s − 1.98·73-s − 1.91·79-s − 5.24·91-s − 1.42·97-s − 1.37·103-s − 0.191·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 6.93·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.313505425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.313505425\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.68474007339944962749448773433, −10.64629209834504164122068357952, −9.597681772198807899677412653505, −9.531899008666477722016168516955, −8.800032022198911717833949569617, −8.607142089093781397251303625625, −7.79072969924423516048558821470, −7.73529321176240727748756940645, −7.32098649877668305079184523340, −7.25140502158361814858796396255, −5.94290793125298037350247414372, −5.58545930878387939553856047688, −5.17408386736887837476315613948, −4.81684582862015619294128793232, −4.33426593469222670368317764669, −4.01417680841473764695521307784, −2.57980856647541561766782447009, −2.56756070212563042614493454616, −1.40988922715689211484239282205, −1.24114364036567459768051279029,
1.24114364036567459768051279029, 1.40988922715689211484239282205, 2.56756070212563042614493454616, 2.57980856647541561766782447009, 4.01417680841473764695521307784, 4.33426593469222670368317764669, 4.81684582862015619294128793232, 5.17408386736887837476315613948, 5.58545930878387939553856047688, 5.94290793125298037350247414372, 7.25140502158361814858796396255, 7.32098649877668305079184523340, 7.73529321176240727748756940645, 7.79072969924423516048558821470, 8.607142089093781397251303625625, 8.800032022198911717833949569617, 9.531899008666477722016168516955, 9.597681772198807899677412653505, 10.64629209834504164122068357952, 10.68474007339944962749448773433
