L(s) = 1 | + 2·9-s − 8·13-s + 4·25-s + 24·37-s + 20·49-s − 8·61-s − 24·73-s − 5·81-s + 40·97-s − 72·109-s − 16·117-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 2/3·9-s − 2.21·13-s + 4/5·25-s + 3.94·37-s + 20/7·49-s − 1.02·61-s − 2.80·73-s − 5/9·81-s + 4.06·97-s − 6.89·109-s − 1.47·117-s − 2.54·121-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8999932237\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8999932237\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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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
−10.52717864377721696717650463318, −10.00185824865765507049710159034, −9.803896087666515595975248140160, −9.381182774179152573794317525580, −9.344325113368977012383708582249, −9.028884395209559184777068333487, −8.719355882125173065570738118366, −8.040588694976626872974073314771, −8.011798526014807660749216778396, −7.62665210053119989042662015850, −7.37939068796191089832665455699, −7.07068135431399251444730317388, −6.87619564194914448952981399519, −6.23029303080468954654585221096, −6.16606299381823662093410340346, −5.48336010762492670936296450623, −5.44805557598716003682004686246, −4.76869937741578979552006429528, −4.48180512233011040460855361201, −4.29881238783314691037563947401, −3.85005515788955566299311392886, −3.01246957505328131734500145666, −2.52327980011448913469364768265, −2.45743383473735650058526571696, −1.23221756643219575807517130617,
1.23221756643219575807517130617, 2.45743383473735650058526571696, 2.52327980011448913469364768265, 3.01246957505328131734500145666, 3.85005515788955566299311392886, 4.29881238783314691037563947401, 4.48180512233011040460855361201, 4.76869937741578979552006429528, 5.44805557598716003682004686246, 5.48336010762492670936296450623, 6.16606299381823662093410340346, 6.23029303080468954654585221096, 6.87619564194914448952981399519, 7.07068135431399251444730317388, 7.37939068796191089832665455699, 7.62665210053119989042662015850, 8.011798526014807660749216778396, 8.040588694976626872974073314771, 8.719355882125173065570738118366, 9.028884395209559184777068333487, 9.344325113368977012383708582249, 9.381182774179152573794317525580, 9.803896087666515595975248140160, 10.00185824865765507049710159034, 10.52717864377721696717650463318