| L(s) = 1 | + 6·3-s + 4·5-s + 4·7-s + 27·9-s − 28·11-s + 26·13-s + 24·15-s − 36·17-s − 44·19-s + 24·21-s − 8·23-s − 226·25-s + 108·27-s + 204·29-s − 164·31-s − 168·33-s + 16·35-s + 668·37-s + 156·39-s − 100·41-s + 272·43-s + 108·45-s + 60·47-s − 566·49-s − 216·51-s + 708·53-s − 112·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.357·5-s + 0.215·7-s + 9-s − 0.767·11-s + 0.554·13-s + 0.413·15-s − 0.513·17-s − 0.531·19-s + 0.249·21-s − 0.0725·23-s − 1.80·25-s + 0.769·27-s + 1.30·29-s − 0.950·31-s − 0.886·33-s + 0.0772·35-s + 2.96·37-s + 0.640·39-s − 0.380·41-s + 0.964·43-s + 0.357·45-s + 0.186·47-s − 1.65·49-s − 0.593·51-s + 1.83·53-s − 0.274·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.090009529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.090009529\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 242 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 582 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 28 T + 2426 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 36 T + 2038 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 44 T - 498 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 8798 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 204 T + 52270 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 164 T + 66294 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 668 T + 212814 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T - 2230 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 272 T + 137142 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 60 T + 203746 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 708 T + 312478 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 180 T + 395626 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1068 T + 726830 T^{2} - 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 420 T + 207854 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 436 T + 749474 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 412 T + 163398 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 672 T + 559646 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 124 T + 501530 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 140 T + 1088138 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 188 T - 343530 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.524309274624230805264607036282, −8.384737634409435292227627961647, −8.063548152737621392501419961613, −7.87055012993826525933335667806, −7.19935052174956647959674031199, −7.04763235369882928778174309826, −6.31932603594109436567061867419, −6.24702410995259432070700037514, −5.54859452769672366399960705513, −5.36904763434846416417056272185, −4.69369724380335325325428054021, −4.21325825265287246956331878583, −3.98827373805609284950387952220, −3.56428371107539416660000196496, −2.84749136830631466327021319422, −2.47584528629301936891550709142, −2.20327455275982344188866013543, −1.69715724988185866733820144001, −0.964627573594920973964316656421, −0.47674032643355995355943430780,
0.47674032643355995355943430780, 0.964627573594920973964316656421, 1.69715724988185866733820144001, 2.20327455275982344188866013543, 2.47584528629301936891550709142, 2.84749136830631466327021319422, 3.56428371107539416660000196496, 3.98827373805609284950387952220, 4.21325825265287246956331878583, 4.69369724380335325325428054021, 5.36904763434846416417056272185, 5.54859452769672366399960705513, 6.24702410995259432070700037514, 6.31932603594109436567061867419, 7.04763235369882928778174309826, 7.19935052174956647959674031199, 7.87055012993826525933335667806, 8.063548152737621392501419961613, 8.384737634409435292227627961647, 8.524309274624230805264607036282
