| L(s) = 1 | + 4·4-s + 20·7-s + 12·16-s + 94·25-s + 80·28-s + 100·37-s + 172·43-s + 202·49-s + 32·64-s − 136·67-s + 20·79-s + 376·100-s − 284·109-s + 240·112-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 400·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 484·169-s + 688·172-s + ⋯ |
| L(s) = 1 | + 4-s + 20/7·7-s + 3/4·16-s + 3.75·25-s + 20/7·28-s + 2.70·37-s + 4·43-s + 4.12·49-s + 1/2·64-s − 2.02·67-s + 0.253·79-s + 3.75·100-s − 2.60·109-s + 15/7·112-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.70·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.86·169-s + 4·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.86517371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.86517371\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 242 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 575 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 698 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 986 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1610 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1058 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 1487 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 4343 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6505 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 5906 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 286 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 9482 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 8231 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 14038 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.014517614612069741855214581933, −7.73341401352917138986315037595, −7.48127233846221047602064385579, −7.34509636476825266265494201498, −7.29605495230952249108718801920, −6.61866987092917672232440447155, −6.47510036392566021615556283731, −6.37975582064446779876857282737, −6.05393961088959237827822187091, −5.50119824820064036242078608352, −5.33663055068141688363414634397, −5.22196160909151239523275262692, −4.98631435157891254892680593249, −4.38725556188966328683668530420, −4.37263075396031592127093201734, −4.17597135031454561648609729539, −3.86218913424866060178895970236, −3.03254102370630186073618284579, −2.79430619425497966326458283290, −2.63897021614654983355718845363, −2.40444031646994145535898793523, −1.77536247402940674600854858470, −1.32557814390095143252893147501, −1.10759612481249533780281477258, −0.795602173409477214538773500995,
0.795602173409477214538773500995, 1.10759612481249533780281477258, 1.32557814390095143252893147501, 1.77536247402940674600854858470, 2.40444031646994145535898793523, 2.63897021614654983355718845363, 2.79430619425497966326458283290, 3.03254102370630186073618284579, 3.86218913424866060178895970236, 4.17597135031454561648609729539, 4.37263075396031592127093201734, 4.38725556188966328683668530420, 4.98631435157891254892680593249, 5.22196160909151239523275262692, 5.33663055068141688363414634397, 5.50119824820064036242078608352, 6.05393961088959237827822187091, 6.37975582064446779876857282737, 6.47510036392566021615556283731, 6.61866987092917672232440447155, 7.29605495230952249108718801920, 7.34509636476825266265494201498, 7.48127233846221047602064385579, 7.73341401352917138986315037595, 8.014517614612069741855214581933
