| L(s) = 1 | − 94·3-s + 330·5-s + 686·7-s + 3.11e3·9-s − 2.84e3·11-s + 2.53e3·13-s − 3.10e4·15-s − 1.48e3·17-s − 3.28e4·19-s − 6.44e4·21-s + 6.57e3·23-s − 5.29e4·25-s + 3.88e4·27-s + 2.06e4·29-s + 3.91e5·31-s + 2.67e5·33-s + 2.26e5·35-s + 3.67e5·37-s − 2.38e5·39-s + 7.34e5·41-s + 4.80e5·43-s + 1.02e6·45-s + 1.08e6·47-s + 3.52e5·49-s + 1.39e5·51-s + 2.85e6·53-s − 9.38e5·55-s + ⋯ |
| L(s) = 1 | − 2.01·3-s + 1.18·5-s + 0.755·7-s + 1.42·9-s − 0.644·11-s + 0.319·13-s − 2.37·15-s − 0.0734·17-s − 1.09·19-s − 1.51·21-s + 0.112·23-s − 0.677·25-s + 0.379·27-s + 0.157·29-s + 2.36·31-s + 1.29·33-s + 0.892·35-s + 1.19·37-s − 0.642·39-s + 1.66·41-s + 0.921·43-s + 1.68·45-s + 1.53·47-s + 3/7·49-s + 0.147·51-s + 2.63·53-s − 0.760·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.059042420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.059042420\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 94 T + 1906 p T^{2} + 94 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 66 p T + 6474 p^{2} T^{2} - 66 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2844 T + 38086566 T^{2} + 2844 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2534 T - 41123742 T^{2} - 2534 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1488 T + 798529822 T^{2} + 1488 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 32810 T + 1897672038 T^{2} + 32810 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6576 T + 6819963598 T^{2} - 6576 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 20640 T + 15579628518 T^{2} - 20640 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 391836 T + 92048864606 T^{2} - 391836 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 367392 T + 63852768182 T^{2} - 367392 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 734664 T + 402811824126 T^{2} - 734664 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 480476 T + 594501933318 T^{2} - 480476 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1089108 T + 1015337137342 T^{2} - 1089108 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2858844 T + 4386858062398 T^{2} - 2858844 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 160170 T + 4361928868198 T^{2} + 160170 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 864646 T + 5755969170906 T^{2} + 864646 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 328648 T + 11587546356582 T^{2} - 328648 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7500216 T + 28549732695406 T^{2} - 7500216 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4301244 T + 18754109784038 T^{2} - 4301244 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6408440 T + 32072611946718 T^{2} - 6408440 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11659074 T + 84453675852838 T^{2} + 11659074 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9772260 T + 83812995056598 T^{2} - 9772260 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10762752 T + 188617737573662 T^{2} - 10762752 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−12.14401563404376708354435627767, −12.12372689629131851263162132636, −11.24125077423467670268656154699, −11.11426162284293672747919933990, −10.44956907097745675998549167952, −10.21813966160504511051112759611, −9.474376063903101579927523494225, −8.802180232469237804919145171250, −8.098460518567572744197640371150, −7.55731553313541047318464472184, −6.43033528490283179805908285415, −6.34570540181904122595316319918, −5.56235607528532353328555802130, −5.50166476864049570075215421650, −4.59590176945231257005460880193, −4.10938612624907684124323519826, −2.50266755430212867268855119345, −2.19382413669127236258518591799, −0.793365079276783378902643368076, −0.74112423165592559827598043713,
0.74112423165592559827598043713, 0.793365079276783378902643368076, 2.19382413669127236258518591799, 2.50266755430212867268855119345, 4.10938612624907684124323519826, 4.59590176945231257005460880193, 5.50166476864049570075215421650, 5.56235607528532353328555802130, 6.34570540181904122595316319918, 6.43033528490283179805908285415, 7.55731553313541047318464472184, 8.098460518567572744197640371150, 8.802180232469237804919145171250, 9.474376063903101579927523494225, 10.21813966160504511051112759611, 10.44956907097745675998549167952, 11.11426162284293672747919933990, 11.24125077423467670268656154699, 12.12372689629131851263162132636, 12.14401563404376708354435627767
