| L(s) = 1 | + 2·9-s − 32·23-s − 4·25-s + 12·49-s − 32·71-s − 40·73-s − 5·81-s − 24·97-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 64·207-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 6.67·23-s − 4/5·25-s + 12/7·49-s − 3.79·71-s − 4.68·73-s − 5/9·81-s − 2.43·97-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-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 + 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s − 4.44·207-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5343293792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5343293792\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.265909017699831400638512889083, −7.947766604517057122997277895903, −7.58377505561171798852967983053, −7.51671614009728113911271684319, −7.42819174845316496691319549424, −7.13901766699093494263261841770, −6.51420947557279667385486354534, −6.49874924579374396926734622456, −5.99830149779478165528715613354, −5.97878864229280295301841996051, −5.75888505131490339091231454492, −5.55266856522322212854624847045, −5.30570611399189772799361049528, −4.48105143546737670939677393280, −4.31013749661651041595736814992, −4.25126178719914202565305768732, −4.09543316838642553122278100661, −3.86827606007007513778805288423, −3.17912207350930069820924720373, −3.00006713872455957799728131876, −2.49080542961897640025428532955, −1.97827223360558049360806343969, −1.73879449695613555507946971694, −1.59836086660238128205971314312, −0.26711616371200848239957197239,
0.26711616371200848239957197239, 1.59836086660238128205971314312, 1.73879449695613555507946971694, 1.97827223360558049360806343969, 2.49080542961897640025428532955, 3.00006713872455957799728131876, 3.17912207350930069820924720373, 3.86827606007007513778805288423, 4.09543316838642553122278100661, 4.25126178719914202565305768732, 4.31013749661651041595736814992, 4.48105143546737670939677393280, 5.30570611399189772799361049528, 5.55266856522322212854624847045, 5.75888505131490339091231454492, 5.97878864229280295301841996051, 5.99830149779478165528715613354, 6.49874924579374396926734622456, 6.51420947557279667385486354534, 7.13901766699093494263261841770, 7.42819174845316496691319549424, 7.51671614009728113911271684319, 7.58377505561171798852967983053, 7.947766604517057122997277895903, 8.265909017699831400638512889083
