| L(s) = 1 | − 7·4-s − 50·9-s + 90·11-s + 64·16-s − 242·19-s + 648·29-s + 176·31-s + 350·36-s + 780·41-s − 630·44-s − 98·49-s − 720·59-s − 784·61-s − 1.00e3·64-s + 192·71-s + 1.69e3·76-s + 1.56e3·79-s + 729·81-s − 2.38e3·89-s − 4.50e3·99-s − 1.36e3·101-s − 3.20e3·109-s − 4.53e3·116-s + 4.68e3·121-s − 1.23e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 7/8·4-s − 1.85·9-s + 2.46·11-s + 16-s − 2.92·19-s + 4.14·29-s + 1.01·31-s + 1.62·36-s + 2.97·41-s − 2.15·44-s − 2/7·49-s − 1.58·59-s − 1.64·61-s − 1.95·64-s + 0.320·71-s + 2.55·76-s + 2.22·79-s + 81-s − 2.84·89-s − 4.56·99-s − 1.34·101-s − 2.81·109-s − 3.63·116-s + 3.52·121-s − 0.892·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.020289732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020289732\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 7 T^{2} - 15 T^{4} + 7 p^{6} T^{6} + p^{12} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 50 T^{2} + 1771 T^{4} + 50 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 45 T + 694 T^{2} - 45 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 913 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 6910 T^{2} + 23610531 T^{4} + 6910 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 121 T + 7782 T^{2} + 121 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 p^{2} T^{2} + 840 p^{4} T^{4} + 37 p^{8} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 162 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 88 T - 22047 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 p^{2} T^{2} - 744 p^{4} T^{4} + 25 p^{8} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 195 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 77218 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 205621 T^{2} + 31500780312 T^{4} + 205621 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 58655 T^{2} - 18723952104 T^{4} - 58655 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 360 T - 75779 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 392 T - 73317 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 523126 T^{2} + 183202429707 T^{4} + 523126 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 331810 T^{2} - 41236350189 T^{4} + 331810 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 782 T + 118485 T^{2} - 782 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 553750 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 1194 T + 720667 T^{2} + 1194 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 1011742 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−8.773496252259794677035414696995, −8.435769808133281364593774830323, −8.372026367714232313728697917052, −8.346602135651939401190294816852, −7.911363412915829001598300777501, −7.59963525531400141588979134749, −6.99405300601949327074166473856, −6.76872735471695772447324712228, −6.34902825764691779396209353506, −6.26672639852028589878260860366, −6.15871435672187466745397182189, −5.92010952786399270230836772125, −5.31632697852662686438668656609, −4.96565855057008544004817519088, −4.58430884525471169418616116709, −4.21267453717774901947659681785, −4.10908515670187555169224368391, −4.03274454405408181039147476193, −2.98814959849854269405432159376, −2.85057492030130992935031571010, −2.84725300216554685714252044493, −1.95308779472424867226352816426, −1.37584895120259891809060741339, −0.870380822616453669437751509887, −0.39046274048992051658667993163,
0.39046274048992051658667993163, 0.870380822616453669437751509887, 1.37584895120259891809060741339, 1.95308779472424867226352816426, 2.84725300216554685714252044493, 2.85057492030130992935031571010, 2.98814959849854269405432159376, 4.03274454405408181039147476193, 4.10908515670187555169224368391, 4.21267453717774901947659681785, 4.58430884525471169418616116709, 4.96565855057008544004817519088, 5.31632697852662686438668656609, 5.92010952786399270230836772125, 6.15871435672187466745397182189, 6.26672639852028589878260860366, 6.34902825764691779396209353506, 6.76872735471695772447324712228, 6.99405300601949327074166473856, 7.59963525531400141588979134749, 7.911363412915829001598300777501, 8.346602135651939401190294816852, 8.372026367714232313728697917052, 8.435769808133281364593774830323, 8.773496252259794677035414696995
