L(s) = 1 | + 4·4-s − 104·29-s − 232·41-s − 100·49-s + 104·61-s − 64·64-s + 328·89-s + 296·101-s + 184·109-s − 416·116-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 928·164-s + 167-s + 668·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 4-s − 3.58·29-s − 5.65·41-s − 2.04·49-s + 1.70·61-s − 64-s + 3.68·89-s + 2.93·101-s + 1.68·109-s − 3.58·116-s + 3.20·121-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 − 5.65·164-s + 0.00598·167-s + 3.95·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.606669560\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.606669560\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 1346 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1150 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8930 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 8542 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 1390 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 11426 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 18814 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.98902686292020002000857447269, −6.77057692349209971531797298310, −6.61235764229200154912364752045, −6.53799204133396654300458880770, −6.03708457365699409013626165805, −6.02998921227114954906906025667, −5.73883972839281606697989246525, −5.36053908067135326217583273304, −5.15498754846501660428307704866, −4.98004958583509001448077390301, −4.84839689986273981628743335308, −4.53592494684129570174402725016, −4.03639584380994739206567476718, −3.85637789603428651709051236909, −3.54785497384153573781015736963, −3.25583913829660784592173995770, −3.13459602729980929185372989468, −3.11376909327485286723188803990, −2.16410854495355785943872730592, −2.03326989321967257497185454359, −1.93808391154810705231376904164, −1.80078905411027921166270814184, −1.29858074920569393959181448174, −0.54414226301140917410685528968, −0.30232173369696205475541553934,
0.30232173369696205475541553934, 0.54414226301140917410685528968, 1.29858074920569393959181448174, 1.80078905411027921166270814184, 1.93808391154810705231376904164, 2.03326989321967257497185454359, 2.16410854495355785943872730592, 3.11376909327485286723188803990, 3.13459602729980929185372989468, 3.25583913829660784592173995770, 3.54785497384153573781015736963, 3.85637789603428651709051236909, 4.03639584380994739206567476718, 4.53592494684129570174402725016, 4.84839689986273981628743335308, 4.98004958583509001448077390301, 5.15498754846501660428307704866, 5.36053908067135326217583273304, 5.73883972839281606697989246525, 6.02998921227114954906906025667, 6.03708457365699409013626165805, 6.53799204133396654300458880770, 6.61235764229200154912364752045, 6.77057692349209971531797298310, 6.98902686292020002000857447269