L(s) = 1 | + 8·13-s − 8·37-s + 16·49-s + 32·61-s + 56·73-s − 9·81-s − 40·97-s + 16·109-s + 4·121-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 + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 2.21·13-s − 1.31·37-s + 16/7·49-s + 4.09·61-s + 6.55·73-s − 81-s − 4.06·97-s + 1.53·109-s + 4/11·121-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 + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.856372879\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.856372879\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−7.00776086012816722965923251691, −6.76978532143042226733252310235, −6.55757142543924919475344300470, −6.25432680947147746309612162174, −6.24768064334008538503096196507, −5.76849217771663929498867791674, −5.57051271290936632965492023754, −5.48198343971334385019764684463, −5.20626979728758973302587263313, −5.03121135831868794505126131594, −4.85438075173400318834763113047, −4.27796491866144545188912360587, −4.01146614273138587152380748832, −3.84662769459138831875254747179, −3.80049165169194238548802076759, −3.73689878620226175120083138195, −3.08696525527140770893926355734, −2.88596053676174090621398366851, −2.76628894480770565724675399282, −2.03004059375596341958878570756, −2.00752560891054778093748432098, −1.87429084522762548668632244192, −1.02265928121029032497586413324, −0.953227584013924735430808775520, −0.57306524434504561419056415933,
0.57306524434504561419056415933, 0.953227584013924735430808775520, 1.02265928121029032497586413324, 1.87429084522762548668632244192, 2.00752560891054778093748432098, 2.03004059375596341958878570756, 2.76628894480770565724675399282, 2.88596053676174090621398366851, 3.08696525527140770893926355734, 3.73689878620226175120083138195, 3.80049165169194238548802076759, 3.84662769459138831875254747179, 4.01146614273138587152380748832, 4.27796491866144545188912360587, 4.85438075173400318834763113047, 5.03121135831868794505126131594, 5.20626979728758973302587263313, 5.48198343971334385019764684463, 5.57051271290936632965492023754, 5.76849217771663929498867791674, 6.24768064334008538503096196507, 6.25432680947147746309612162174, 6.55757142543924919475344300470, 6.76978532143042226733252310235, 7.00776086012816722965923251691