| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 9-s + 6·13-s + 5·16-s + 2·18-s − 12·26-s + 12·29-s − 6·32-s − 3·36-s + 12·37-s − 14·49-s + 18·52-s − 24·58-s + 20·61-s + 7·64-s + 4·72-s − 12·73-s − 24·74-s + 16·79-s + 81-s + 12·97-s + 28·98-s + 36·101-s − 24·104-s + 36·116-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 1/3·9-s + 1.66·13-s + 5/4·16-s + 0.471·18-s − 2.35·26-s + 2.22·29-s − 1.06·32-s − 1/2·36-s + 1.97·37-s − 2·49-s + 2.49·52-s − 3.15·58-s + 2.56·61-s + 7/8·64-s + 0.471·72-s − 1.40·73-s − 2.78·74-s + 1.80·79-s + 1/9·81-s + 1.21·97-s + 2.82·98-s + 3.58·101-s − 2.35·104-s + 3.34·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.528717282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.528717282\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.315564361703423866839312813852, −8.915342546585877387584698931824, −8.571607450048986613797966848657, −8.290875676233602380852611979344, −7.972392302053114200047912463317, −7.69987883288446954058786238490, −6.93764171522619575985984597810, −6.77666953726602891367454982538, −6.24076242744103155978375811724, −6.04741809293893374284449214887, −5.64646561721290103595272044341, −4.87371172908589703045445020149, −4.56020683791986381403998741943, −3.90184521306070144586832276102, −3.25440763180800176213597376437, −3.07563840814436985118267793554, −2.30802308264166017801586913451, −1.85624990556623774220108223634, −0.971702162256151487995869343224, −0.74325471506149291537588844980,
0.74325471506149291537588844980, 0.971702162256151487995869343224, 1.85624990556623774220108223634, 2.30802308264166017801586913451, 3.07563840814436985118267793554, 3.25440763180800176213597376437, 3.90184521306070144586832276102, 4.56020683791986381403998741943, 4.87371172908589703045445020149, 5.64646561721290103595272044341, 6.04741809293893374284449214887, 6.24076242744103155978375811724, 6.77666953726602891367454982538, 6.93764171522619575985984597810, 7.69987883288446954058786238490, 7.972392302053114200047912463317, 8.290875676233602380852611979344, 8.571607450048986613797966848657, 8.915342546585877387584698931824, 9.315564361703423866839312813852
