| L(s) = 1 | − 16-s − 4·19-s − 4·49-s − 4·61-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 16-s − 4·19-s − 4·49-s − 4·61-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1897071349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1897071349\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.24597591718127930628764576941, −6.10727422809373159310536179858, −6.00477529917657942880727165907, −5.87879829589252799161627287048, −5.33182694746686031333768168516, −5.17550548795675326349954246620, −4.98217247665878702418684643061, −4.64682768377721411346024987088, −4.61350983078824726940189166123, −4.57505655808519907944404620231, −4.24353879066118819682560675171, −4.14603874487716110791227026129, −3.78500489946826328544568450396, −3.49023799246816021736920406591, −3.44094805437287590450927038824, −2.98954593424674399346939061585, −2.96059460854360747427595305849, −2.60723553073464479934983299064, −2.30813523539259161066549742251, −2.11342649451809806666003523781, −1.81600961662093426441045057405, −1.61437694517079147150296135734, −1.56873612633801131118794968377, −0.864490264831248719013907376912, −0.16671518267732937658998879965,
0.16671518267732937658998879965, 0.864490264831248719013907376912, 1.56873612633801131118794968377, 1.61437694517079147150296135734, 1.81600961662093426441045057405, 2.11342649451809806666003523781, 2.30813523539259161066549742251, 2.60723553073464479934983299064, 2.96059460854360747427595305849, 2.98954593424674399346939061585, 3.44094805437287590450927038824, 3.49023799246816021736920406591, 3.78500489946826328544568450396, 4.14603874487716110791227026129, 4.24353879066118819682560675171, 4.57505655808519907944404620231, 4.61350983078824726940189166123, 4.64682768377721411346024987088, 4.98217247665878702418684643061, 5.17550548795675326349954246620, 5.33182694746686031333768168516, 5.87879829589252799161627287048, 6.00477529917657942880727165907, 6.10727422809373159310536179858, 6.24597591718127930628764576941
