| L(s) = 1 | − 16·13-s + 64·37-s + 76·49-s − 232·61-s − 376·73-s + 56·97-s + 88·109-s − 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 516·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 1.23·13-s + 1.72·37-s + 1.55·49-s − 3.80·61-s − 5.15·73-s + 0.577·97-s + 0.807·109-s − 0.958·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.05·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05572572389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05572572389\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 398 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1682 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 3218 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3202 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6662 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 6818 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 8882 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12242 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 2978 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 15122 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
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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
−5.88883156781217146628793407386, −5.87048829913370696413881057641, −5.58816091413272510007317180581, −5.17738576360351542924825310578, −4.94877951507534115323727189648, −4.87991811148076673686564777994, −4.71306090139151421199988471581, −4.32758709565971781093589677109, −4.31862935192948002840353424197, −4.14115771347557592368729817295, −4.02845582650908554581326878998, −3.40239620690260923865561749429, −3.20602297105433413918829899668, −3.17050288575851118336582904522, −3.15940970905270758914320749256, −2.51921749180829626638658444002, −2.42283412925148344249802783069, −2.22371445119751370551723432436, −2.22305364956983887992961055106, −1.52572348781683388555351994889, −1.28928260715697753141149271080, −1.27505708092647970103911201011, −0.946760044687220483791146840163, −0.33697009826577468473852504358, −0.03737945455787222118377945466,
0.03737945455787222118377945466, 0.33697009826577468473852504358, 0.946760044687220483791146840163, 1.27505708092647970103911201011, 1.28928260715697753141149271080, 1.52572348781683388555351994889, 2.22305364956983887992961055106, 2.22371445119751370551723432436, 2.42283412925148344249802783069, 2.51921749180829626638658444002, 3.15940970905270758914320749256, 3.17050288575851118336582904522, 3.20602297105433413918829899668, 3.40239620690260923865561749429, 4.02845582650908554581326878998, 4.14115771347557592368729817295, 4.31862935192948002840353424197, 4.32758709565971781093589677109, 4.71306090139151421199988471581, 4.87991811148076673686564777994, 4.94877951507534115323727189648, 5.17738576360351542924825310578, 5.58816091413272510007317180581, 5.87048829913370696413881057641, 5.88883156781217146628793407386
