| L(s) = 1 | − 68·11-s − 8·19-s + 92·29-s − 520·31-s + 96·41-s + 682·49-s + 4·59-s − 76·61-s + 1.41e3·71-s + 1.70e3·79-s + 2.76e3·89-s − 1.40e3·101-s − 1.20e3·109-s + 806·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 230·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.86·11-s − 0.0965·19-s + 0.589·29-s − 3.01·31-s + 0.365·41-s + 1.98·49-s + 0.00882·59-s − 0.159·61-s + 2.36·71-s + 2.42·79-s + 3.28·89-s − 1.38·101-s − 1.05·109-s + 0.605·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.104·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.453817472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453817472\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 230 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1230 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 260 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 3962 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 119014 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 196830 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 541990 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 708 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 635150 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 852 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 431238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1380 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1561150 T^{2} + p^{6} T^{4} \) |
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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
−9.035908666784691999736101897209, −8.922013161679591478974999430278, −8.037386346182623124144825384861, −8.017666463992369804593528506934, −7.61692977063714902889143877774, −7.26843982751218006092893734642, −6.60132306249740675139085559231, −6.53142425400459819007250692909, −5.60488737358957664894247925099, −5.50623895790388903831508603129, −5.19386388806185095956280555275, −4.72836856464709470342183853592, −3.95417737132446741696620288424, −3.82260309886258355834662428507, −3.09068205238293067528554307534, −2.71120859256840292044141119559, −1.97739074917696641986942044058, −1.93768075108107013089919700456, −0.69633812930821871988978238510, −0.47472698697582435819906859719,
0.47472698697582435819906859719, 0.69633812930821871988978238510, 1.93768075108107013089919700456, 1.97739074917696641986942044058, 2.71120859256840292044141119559, 3.09068205238293067528554307534, 3.82260309886258355834662428507, 3.95417737132446741696620288424, 4.72836856464709470342183853592, 5.19386388806185095956280555275, 5.50623895790388903831508603129, 5.60488737358957664894247925099, 6.53142425400459819007250692909, 6.60132306249740675139085559231, 7.26843982751218006092893734642, 7.61692977063714902889143877774, 8.017666463992369804593528506934, 8.037386346182623124144825384861, 8.922013161679591478974999430278, 9.035908666784691999736101897209
