| L(s) = 1 | − 48·19-s + 42·25-s + 152·31-s + 104·49-s − 280·61-s − 120·79-s + 296·109-s + 288·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 152·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 2.52·19-s + 1.67·25-s + 4.90·31-s + 2.12·49-s − 4.59·61-s − 1.51·79-s + 2.71·109-s + 2.38·121-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 − 0.899·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.214707337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214707337\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 42 T^{2} + p^{4} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 52 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 144 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 42 p T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1610 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2692 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1440 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 902 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1470 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 2798 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 4030 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 10474 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 4254 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14784 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 9802 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.36052330904968145794517324660, −6.90629747753969979542327658630, −6.83708201655075239321110343827, −6.51063338271153183142230852327, −6.31684737887153408583525080951, −6.09555595944207973957427994680, −5.97881284645072039736016796429, −5.85056945671979228775462136859, −5.35134387527944707559252729274, −4.90483348621287154714825787387, −4.74751397708240029935046010874, −4.54452813832720222889711051919, −4.45160594224813587226764565586, −4.28724527023839170645223133491, −3.90377745000679921489198038403, −3.28384711569160241651586701261, −3.23959928664010922161579251620, −2.98998293888909221110874308288, −2.40863458462147401742208820245, −2.38219398738874057388919931427, −2.22131690238259268019193647486, −1.32603783672534691073634855723, −1.24464943366062013008960699521, −0.862865440054364603715582903708, −0.18996158735494149256740326497,
0.18996158735494149256740326497, 0.862865440054364603715582903708, 1.24464943366062013008960699521, 1.32603783672534691073634855723, 2.22131690238259268019193647486, 2.38219398738874057388919931427, 2.40863458462147401742208820245, 2.98998293888909221110874308288, 3.23959928664010922161579251620, 3.28384711569160241651586701261, 3.90377745000679921489198038403, 4.28724527023839170645223133491, 4.45160594224813587226764565586, 4.54452813832720222889711051919, 4.74751397708240029935046010874, 4.90483348621287154714825787387, 5.35134387527944707559252729274, 5.85056945671979228775462136859, 5.97881284645072039736016796429, 6.09555595944207973957427994680, 6.31684737887153408583525080951, 6.51063338271153183142230852327, 6.83708201655075239321110343827, 6.90629747753969979542327658630, 7.36052330904968145794517324660
