| L(s) = 1 | − 8·4-s + 48·16-s + 50·25-s + 146·37-s + 122·43-s − 256·64-s − 26·67-s + 22·79-s − 400·100-s − 142·109-s − 242·121-s + 127-s + 131-s + 137-s + 139-s − 1.16e3·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 191·169-s − 976·172-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 2·25-s + 3.94·37-s + 2.83·43-s − 4·64-s − 0.388·67-s + 0.278·79-s − 4·100-s − 1.30·109-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7.89·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.13·169-s − 5.67·172-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.493726169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493726169\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )( 1 + 74 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
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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
−10.89707641539154980596377826453, −10.73439648968899534058696743640, −10.15524653090223119618197443881, −9.540460978679954416689709443905, −9.231667596392539947081483964347, −9.178135075269650897683331185780, −8.312183899720989695300846506065, −8.191222601684563690203151960256, −7.57158796198353504656296830374, −7.14395071079933711743176236433, −6.10856293450598234469053637259, −6.02678892702023986165466494787, −5.25306688999071825885957542209, −4.80555258545325935908296396160, −4.16377038511814639966701223946, −4.12760894767927516786191289460, −3.06943119549509153015561609338, −2.60225555093138646316708067837, −1.09235090126459240591045842419, −0.67987782170374530752488418708,
0.67987782170374530752488418708, 1.09235090126459240591045842419, 2.60225555093138646316708067837, 3.06943119549509153015561609338, 4.12760894767927516786191289460, 4.16377038511814639966701223946, 4.80555258545325935908296396160, 5.25306688999071825885957542209, 6.02678892702023986165466494787, 6.10856293450598234469053637259, 7.14395071079933711743176236433, 7.57158796198353504656296830374, 8.191222601684563690203151960256, 8.312183899720989695300846506065, 9.178135075269650897683331185780, 9.231667596392539947081483964347, 9.540460978679954416689709443905, 10.15524653090223119618197443881, 10.73439648968899534058696743640, 10.89707641539154980596377826453
