| L(s) = 1 | − 8·13-s − 2·25-s − 40·37-s + 4·49-s − 52·61-s − 32·73-s + 16·97-s − 44·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.21·13-s − 2/5·25-s − 6.57·37-s + 4/7·49-s − 6.65·61-s − 3.74·73-s + 1.62·97-s − 4.21·109-s − 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3977962525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3977962525\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 155 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
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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
−6.42755417029701304426054867357, −6.22919027584046014185510994772, −6.14464544666951906539186245383, −5.72486362226170200388350313186, −5.49250325487413030194522794072, −5.35072344381041053010886322962, −5.30841242151769530143650076893, −4.81760635276687350194482991483, −4.80097642682377154319529095016, −4.70079078957513678525056096218, −4.42946651115364882901762550661, −3.97827788728404836419985525424, −3.85646006439652146406591185466, −3.57374723464529173957064916846, −3.46983503142094616244111368129, −2.92749244958227729216890557208, −2.86198514040098605096558935053, −2.65995036506408856017106654831, −2.58306615156410037731628597632, −1.69686979324372111366877841769, −1.68846449731124957408339572200, −1.61694780517689282226842465619, −1.52672076330851204507637363778, −0.36159135394886704521848194993, −0.19761189563082437227320471824,
0.19761189563082437227320471824, 0.36159135394886704521848194993, 1.52672076330851204507637363778, 1.61694780517689282226842465619, 1.68846449731124957408339572200, 1.69686979324372111366877841769, 2.58306615156410037731628597632, 2.65995036506408856017106654831, 2.86198514040098605096558935053, 2.92749244958227729216890557208, 3.46983503142094616244111368129, 3.57374723464529173957064916846, 3.85646006439652146406591185466, 3.97827788728404836419985525424, 4.42946651115364882901762550661, 4.70079078957513678525056096218, 4.80097642682377154319529095016, 4.81760635276687350194482991483, 5.30841242151769530143650076893, 5.35072344381041053010886322962, 5.49250325487413030194522794072, 5.72486362226170200388350313186, 6.14464544666951906539186245383, 6.22919027584046014185510994772, 6.42755417029701304426054867357
