| L(s) = 1 | − 2·5-s + 5·7-s + 11-s − 2·13-s + 17-s + 2·19-s + 9·23-s + 3·25-s + 6·29-s − 12·31-s − 10·35-s − 9·37-s − 3·41-s + 2·43-s + 14·47-s + 9·49-s − 3·53-s − 2·55-s + 24·59-s − 61-s + 4·65-s − 2·67-s + 25·71-s − 12·73-s + 5·77-s + 3·79-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.88·7-s + 0.301·11-s − 0.554·13-s + 0.242·17-s + 0.458·19-s + 1.87·23-s + 3/5·25-s + 1.11·29-s − 2.15·31-s − 1.69·35-s − 1.47·37-s − 0.468·41-s + 0.304·43-s + 2.04·47-s + 9/7·49-s − 0.412·53-s − 0.269·55-s + 3.12·59-s − 0.128·61-s + 0.496·65-s − 0.244·67-s + 2.96·71-s − 1.40·73-s + 0.569·77-s + 0.337·79-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.281888814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.281888814\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 25 T + 294 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 122 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 160 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T - 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69321788541010938939413965391, −7.54117292379582250177734474009, −7.20846640506495796636931006791, −7.16892460395433082860188502951, −6.48482262254062153372197414145, −6.41575917892151941645790823884, −5.44130959762131830435095843051, −5.36302974769799629566223315223, −5.13894788039934356238748244365, −4.96764697191884060435266796461, −4.23180669863982282444685653767, −4.19470506393046977862654132405, −3.61805728124264033400333108937, −3.40873176592860236043905890097, −2.71379773417269521693085822773, −2.50649728979290284466006458652, −1.71523426037554875366604296300, −1.65049799114661826268679954907, −0.843267493096515654077266000295, −0.60570654296799325185512357845,
0.60570654296799325185512357845, 0.843267493096515654077266000295, 1.65049799114661826268679954907, 1.71523426037554875366604296300, 2.50649728979290284466006458652, 2.71379773417269521693085822773, 3.40873176592860236043905890097, 3.61805728124264033400333108937, 4.19470506393046977862654132405, 4.23180669863982282444685653767, 4.96764697191884060435266796461, 5.13894788039934356238748244365, 5.36302974769799629566223315223, 5.44130959762131830435095843051, 6.41575917892151941645790823884, 6.48482262254062153372197414145, 7.16892460395433082860188502951, 7.20846640506495796636931006791, 7.54117292379582250177734474009, 7.69321788541010938939413965391
