| L(s) = 1 | − 2·5-s − 6·11-s − 12·13-s + 2·17-s − 4·19-s − 2·23-s + 5·25-s + 16·29-s − 4·31-s + 6·37-s + 20·41-s − 8·43-s + 4·47-s + 4·53-s + 12·55-s + 12·59-s + 2·61-s + 24·65-s − 12·67-s + 12·71-s + 2·73-s + 8·79-s − 4·85-s + 14·89-s + 8·95-s − 4·97-s + 18·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.80·11-s − 3.32·13-s + 0.485·17-s − 0.917·19-s − 0.417·23-s + 25-s + 2.97·29-s − 0.718·31-s + 0.986·37-s + 3.12·41-s − 1.21·43-s + 0.583·47-s + 0.549·53-s + 1.61·55-s + 1.56·59-s + 0.256·61-s + 2.97·65-s − 1.46·67-s + 1.42·71-s + 0.234·73-s + 0.900·79-s − 0.433·85-s + 1.48·89-s + 0.820·95-s − 0.406·97-s + 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.351530836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351530836\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.492474415400537979106175373553, −8.403407073208189605002474165311, −7.81835855618017360641556395425, −7.71204472913453961375092288518, −7.30748730506948330074668685119, −7.21127570555540978533765688535, −6.54699529632811106836418218097, −6.21116920818554400341986104158, −5.68188692185819798270310781399, −5.15792271346531491338176029494, −4.81902058869356988402700580510, −4.74657430411269896256017203262, −4.28911323581950568590629415399, −3.74935043955816335674645271539, −2.97584687203881023110663639492, −2.77495405506953136436756080208, −2.25900572574214636740272990688, −2.21809125443337599213833066095, −0.68141682516951546714755150837, −0.54236527448166837249654009028,
0.54236527448166837249654009028, 0.68141682516951546714755150837, 2.21809125443337599213833066095, 2.25900572574214636740272990688, 2.77495405506953136436756080208, 2.97584687203881023110663639492, 3.74935043955816335674645271539, 4.28911323581950568590629415399, 4.74657430411269896256017203262, 4.81902058869356988402700580510, 5.15792271346531491338176029494, 5.68188692185819798270310781399, 6.21116920818554400341986104158, 6.54699529632811106836418218097, 7.21127570555540978533765688535, 7.30748730506948330074668685119, 7.71204472913453961375092288518, 7.81835855618017360641556395425, 8.403407073208189605002474165311, 8.492474415400537979106175373553
