| L(s) = 1 | − 4-s + 9-s + 2·31-s − 36-s − 49-s + 2·59-s − 2·71-s + 8·89-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4-s + 9-s + 2·31-s − 36-s − 49-s + 2·59-s − 2·71-s + 8·89-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 196-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.293139052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.293139052\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 3 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 37 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 59 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$ | \( ( 1 - T )^{8} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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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
−6.38688192114351780713308688419, −6.26215594049809338450300095306, −5.89237501676137510682339954913, −5.75363140822042313008061112701, −5.70452094168977333896231709206, −5.20667232263036746911658833724, −4.99569410438809019900586885819, −4.86516480231054871234342287693, −4.83279588719347990127203577688, −4.50389218712909490844417983709, −4.40127496899301648621571448866, −4.20946782737479113767621070635, −3.90714982114245832496713604655, −3.55680358935366935384333542412, −3.44184515103002527813404887507, −3.42015113556293929284348241979, −3.02747941243955528740200309123, −2.60517708748495607011610822641, −2.40046962897026905977320897310, −2.25816433759928509232252633338, −1.95125099815500254925731718884, −1.54933882242620815637663757873, −1.25715786843295001244008191271, −0.921513808326219850436467154319, −0.57309192233742260054144007607,
0.57309192233742260054144007607, 0.921513808326219850436467154319, 1.25715786843295001244008191271, 1.54933882242620815637663757873, 1.95125099815500254925731718884, 2.25816433759928509232252633338, 2.40046962897026905977320897310, 2.60517708748495607011610822641, 3.02747941243955528740200309123, 3.42015113556293929284348241979, 3.44184515103002527813404887507, 3.55680358935366935384333542412, 3.90714982114245832496713604655, 4.20946782737479113767621070635, 4.40127496899301648621571448866, 4.50389218712909490844417983709, 4.83279588719347990127203577688, 4.86516480231054871234342287693, 4.99569410438809019900586885819, 5.20667232263036746911658833724, 5.70452094168977333896231709206, 5.75363140822042313008061112701, 5.89237501676137510682339954913, 6.26215594049809338450300095306, 6.38688192114351780713308688419
