| L(s) = 1 | + 1.02e3·17-s − 1.08e3·25-s − 2.18e3·41-s + 2.43e3·49-s + 3.48e4·73-s − 1.31e4·81-s − 2.04e3·89-s + 3.07e3·97-s − 2.22e4·113-s + 1.43e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6.13e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.54·17-s − 1.73·25-s − 1.29·41-s + 1.01·49-s + 6.53·73-s − 2·81-s − 0.258·89-s + 0.326·97-s − 1.74·113-s + 0.979·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s − 2.14·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.581645404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.581645404\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p^{8} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 542 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 174 p T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 7168 T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 30686 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 256 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 215040 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 527426 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 294562 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1832706 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 3531490 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 546 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 5476352 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 8325762 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 15564130 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 18529280 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6400930 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 38728704 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 24928962 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8704 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 67449218 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 75758592 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 512 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 768 T + p^{4} T^{2} )^{4} \) |
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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
−7.50331675202032448865098831494, −7.00779135199918831931105408475, −6.71142781149008801806279869765, −6.52273204415444623700983751067, −6.25617792769491598478361428446, −5.86931819907274758643416925971, −5.79197106450791629703427416804, −5.46731968717102736007078207419, −5.29104877281717278475952969087, −5.04729106223262058402905695494, −4.90240614295102960135882908220, −4.41760648533201983144232534828, −3.94898157812738274472059746750, −3.86992517039253216480975965258, −3.50447739320710448195666580349, −3.46442070146040351453743998485, −3.14030805045232980248187916873, −2.56848345739522521876902647617, −2.46110496031461217586366169428, −2.02144430267173323323915299554, −1.63675504634410839258925276917, −1.28504288271290179141582129413, −1.02010324163128514148235460074, −0.63378864331781010939863893717, −0.28390359030261997638382387865,
0.28390359030261997638382387865, 0.63378864331781010939863893717, 1.02010324163128514148235460074, 1.28504288271290179141582129413, 1.63675504634410839258925276917, 2.02144430267173323323915299554, 2.46110496031461217586366169428, 2.56848345739522521876902647617, 3.14030805045232980248187916873, 3.46442070146040351453743998485, 3.50447739320710448195666580349, 3.86992517039253216480975965258, 3.94898157812738274472059746750, 4.41760648533201983144232534828, 4.90240614295102960135882908220, 5.04729106223262058402905695494, 5.29104877281717278475952969087, 5.46731968717102736007078207419, 5.79197106450791629703427416804, 5.86931819907274758643416925971, 6.25617792769491598478361428446, 6.52273204415444623700983751067, 6.71142781149008801806279869765, 7.00779135199918831931105408475, 7.50331675202032448865098831494
