| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 5·9-s − 4·14-s + 16-s − 3·17-s + 5·18-s + 5·23-s − 25-s + 4·28-s − 5·31-s − 32-s + 3·34-s − 5·36-s − 15·41-s − 5·46-s − 3·47-s + 7·49-s + 50-s − 4·56-s + 5·62-s − 20·63-s + 64-s − 3·68-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s − 0.727·17-s + 1.17·18-s + 1.04·23-s − 1/5·25-s + 0.755·28-s − 0.898·31-s − 0.176·32-s + 0.514·34-s − 5/6·36-s − 2.34·41-s − 0.737·46-s − 0.437·47-s + 49-s + 0.141·50-s − 0.534·56-s + 0.635·62-s − 2.51·63-s + 1/8·64-s − 0.363·68-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5594681166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5594681166\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−12.78912282019567903027660297416, −11.83783159581439458116776577470, −11.61427577959493971618095536628, −10.95456420545025768805091152100, −10.74399486704281593022944103063, −9.682467835415954696911097446948, −8.917722921850910464016775230536, −8.493217092368260651723415382850, −8.048385928525224176489037386795, −7.20315367744419609328011967740, −6.38172203634760313512956515328, −5.40491758132966709594557643401, −4.86576179293298387663308554451, −3.35991069196061239692692583694, −2.06813763014627359933851385827,
2.06813763014627359933851385827, 3.35991069196061239692692583694, 4.86576179293298387663308554451, 5.40491758132966709594557643401, 6.38172203634760313512956515328, 7.20315367744419609328011967740, 8.048385928525224176489037386795, 8.493217092368260651723415382850, 8.917722921850910464016775230536, 9.682467835415954696911097446948, 10.74399486704281593022944103063, 10.95456420545025768805091152100, 11.61427577959493971618095536628, 11.83783159581439458116776577470, 12.78912282019567903027660297416
