| L(s) = 1 | − 2·3-s − 5-s + 3·9-s + 4·11-s + 2·13-s + 2·15-s − 7·19-s + 3·23-s − 25-s − 4·27-s − 3·29-s + 31-s − 8·33-s + 2·37-s − 4·39-s − 6·41-s + 5·43-s − 3·45-s + 7·47-s + 5·53-s − 4·55-s + 14·57-s + 16·59-s − 12·61-s − 2·65-s + 14·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.60·19-s + 0.625·23-s − 1/5·25-s − 0.769·27-s − 0.557·29-s + 0.179·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.762·43-s − 0.447·45-s + 1.02·47-s + 0.686·53-s − 0.539·55-s + 1.85·57-s + 2.08·59-s − 1.53·61-s − 0.248·65-s + 1.71·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894418796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894418796\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 98 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T - 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 98 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9 T + 190 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 29 T + 396 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
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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
−7.83422983549892817206374131692, −7.80646403650249034745819303763, −7.17339974202224055847078632513, −6.91374354999532100307991331989, −6.58621968710743022131789556477, −6.45196946005752336279303381949, −5.83439717668421762673464716387, −5.80124821209625027960061926441, −5.25697696782888812571226921188, −4.93447468015548704998185027680, −4.33948800390894330558888376112, −4.23794332031473584731802141805, −3.74456166261019688731733419535, −3.67961576420719775120124934790, −2.87873491698267737217202065508, −2.45659779218722649991064154283, −1.79461396008775756194560689391, −1.54989730858972343540937227116, −0.76092106459037558910641398196, −0.50785677616995654619015389337,
0.50785677616995654619015389337, 0.76092106459037558910641398196, 1.54989730858972343540937227116, 1.79461396008775756194560689391, 2.45659779218722649991064154283, 2.87873491698267737217202065508, 3.67961576420719775120124934790, 3.74456166261019688731733419535, 4.23794332031473584731802141805, 4.33948800390894330558888376112, 4.93447468015548704998185027680, 5.25697696782888812571226921188, 5.80124821209625027960061926441, 5.83439717668421762673464716387, 6.45196946005752336279303381949, 6.58621968710743022131789556477, 6.91374354999532100307991331989, 7.17339974202224055847078632513, 7.80646403650249034745819303763, 7.83422983549892817206374131692
