L(s) = 1 | + 4·3-s + 8·7-s + 4·9-s + 12·19-s + 32·21-s + 12·25-s − 4·27-s − 16·37-s + 34·49-s + 48·57-s + 36·59-s + 32·63-s + 48·75-s − 10·81-s + 36·83-s + 48·103-s − 64·111-s − 24·113-s + 12·121-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + 136·147-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 3.02·7-s + 4/3·9-s + 2.75·19-s + 6.98·21-s + 12/5·25-s − 0.769·27-s − 2.63·37-s + 34/7·49-s + 6.35·57-s + 4.68·59-s + 4.03·63-s + 5.54·75-s − 1.11·81-s + 3.95·83-s + 4.72·103-s − 6.07·111-s − 2.25·113-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + 11.2·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(24.77964951\) |
\(L(\frac12)\) |
\(\approx\) |
\(24.77964951\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
good | 3 | $D_{4}$ | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 426 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3414 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 18 T + 172 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 6186 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 244 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.65127397876537475189678540714, −6.54828033878958475927840619234, −6.39817283727256316719675304594, −5.68269782999461804617552959280, −5.57940187706500157296648458374, −5.36503800484731154417755773162, −5.25055608999607256001843169193, −5.19964954310748762484086287959, −4.94813385345460826016104361156, −4.62132978223779372232419129059, −4.47728641549432196620877133598, −4.18918020746038407995323999605, −3.83583068349452777716266675834, −3.56236640555768573406710884944, −3.24895834688436860532750468647, −3.22002225352805589147018759302, −3.15937721835126699757389096032, −2.64157986983474498769625291627, −2.37961169605677306685158279935, −2.12590284817796909370800595803, −1.93689706775363467564271433827, −1.79939323242850250828922193375, −1.07096604102383221240591265392, −0.996521312305338989560089000490, −0.78831150340855633856001371166,
0.78831150340855633856001371166, 0.996521312305338989560089000490, 1.07096604102383221240591265392, 1.79939323242850250828922193375, 1.93689706775363467564271433827, 2.12590284817796909370800595803, 2.37961169605677306685158279935, 2.64157986983474498769625291627, 3.15937721835126699757389096032, 3.22002225352805589147018759302, 3.24895834688436860532750468647, 3.56236640555768573406710884944, 3.83583068349452777716266675834, 4.18918020746038407995323999605, 4.47728641549432196620877133598, 4.62132978223779372232419129059, 4.94813385345460826016104361156, 5.19964954310748762484086287959, 5.25055608999607256001843169193, 5.36503800484731154417755773162, 5.57940187706500157296648458374, 5.68269782999461804617552959280, 6.39817283727256316719675304594, 6.54828033878958475927840619234, 6.65127397876537475189678540714