| L(s) = 1 | + 2·4-s + 4·13-s + 46·19-s − 32·25-s + 94·31-s + 110·37-s + 92·43-s + 8·52-s + 208·61-s − 8·64-s + 194·67-s + 130·73-s + 92·76-s − 226·79-s − 416·97-s − 64·100-s + 238·103-s + 98·109-s − 80·121-s + 188·124-s + 127-s + 131-s + 137-s + 139-s + 220·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 4/13·13-s + 2.42·19-s − 1.27·25-s + 3.03·31-s + 2.97·37-s + 2.13·43-s + 2/13·52-s + 3.40·61-s − 1/8·64-s + 2.89·67-s + 1.78·73-s + 1.21·76-s − 2.86·79-s − 4.28·97-s − 0.639·100-s + 2.31·103-s + 0.899·109-s − 0.661·121-s + 1.51·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.48·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.31749535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.31749535\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 80 T^{2} - 8241 T^{4} + 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 290 T^{2} + 579 T^{4} + 290 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 23 T + 168 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 47 T + 1248 T^{2} - 47 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 55 T + 1656 T^{2} - 55 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 4400 T^{2} + 14480319 T^{4} + 4400 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 3026 T^{2} + 1266195 T^{4} + 3026 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 82 T + 3243 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} )( 1 + 82 T + 3243 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - 104 T + 7095 T^{2} - 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 97 T + 4920 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 560 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 65 T - 1104 T^{2} - 65 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 113 T + 6528 T^{2} + 113 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12896 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 2590 T^{2} - 56034141 T^{4} - 2590 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 104 T + p^{2} T^{2} )^{4} \) |
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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
−7.11215120167335705718539066712, −6.83204225245243557456870946701, −6.60287619205933761807338443722, −6.41417459122697188194771653742, −6.18254520411938844040178923367, −5.78130473143812503631309748912, −5.62102557353525299232055117426, −5.61323856939952557296600190676, −5.41882174872234355810414217060, −4.79059161416349287125462381075, −4.76748319442098445423032070166, −4.38541112136983324777001557266, −4.28838311458721586253286546476, −3.80738305641596399747718637649, −3.80288116903528382436461741884, −3.27791016337674869306805119521, −3.14950856175465587134293012830, −2.67661858781656140749936850067, −2.46513254758780108580677343838, −2.39321785466991094162215090031, −1.99480493508842046158376806345, −1.30565479749577311215100566947, −1.07883334071412908599988494406, −0.72349979929513840154668891986, −0.62450697325769302954227057067,
0.62450697325769302954227057067, 0.72349979929513840154668891986, 1.07883334071412908599988494406, 1.30565479749577311215100566947, 1.99480493508842046158376806345, 2.39321785466991094162215090031, 2.46513254758780108580677343838, 2.67661858781656140749936850067, 3.14950856175465587134293012830, 3.27791016337674869306805119521, 3.80288116903528382436461741884, 3.80738305641596399747718637649, 4.28838311458721586253286546476, 4.38541112136983324777001557266, 4.76748319442098445423032070166, 4.79059161416349287125462381075, 5.41882174872234355810414217060, 5.61323856939952557296600190676, 5.62102557353525299232055117426, 5.78130473143812503631309748912, 6.18254520411938844040178923367, 6.41417459122697188194771653742, 6.60287619205933761807338443722, 6.83204225245243557456870946701, 7.11215120167335705718539066712
