| L(s) = 1 | + 8·2-s + 14·3-s + 48·4-s + 70·5-s + 112·6-s + 256·8-s − 23·9-s + 560·10-s + 62·11-s + 672·12-s + 1.82e3·13-s + 980·15-s + 1.28e3·16-s + 1.69e3·17-s − 184·18-s + 826·19-s + 3.36e3·20-s + 496·22-s − 2.73e3·23-s + 3.58e3·24-s + 2.48e3·25-s + 1.45e4·26-s + 14·27-s − 2.85e3·29-s + 7.84e3·30-s − 2.67e3·31-s + 6.14e3·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.898·3-s + 3/2·4-s + 1.25·5-s + 1.27·6-s + 1.41·8-s − 0.0946·9-s + 1.77·10-s + 0.154·11-s + 1.34·12-s + 2.98·13-s + 1.12·15-s + 5/4·16-s + 1.42·17-s − 0.133·18-s + 0.524·19-s + 1.87·20-s + 0.218·22-s − 1.07·23-s + 1.27·24-s + 0.793·25-s + 4.22·26-s + 0.00369·27-s − 0.629·29-s + 1.59·30-s − 0.499·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(14.00388775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.00388775\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 73 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 14 p T + 2419 T^{2} - 14 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 62 T + 307579 T^{2} - 62 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 140 p T + 1508750 T^{2} - 140 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1694 T + 3404179 T^{2} - 1694 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 826 T + 4642131 T^{2} - 826 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2734 T + 10266499 T^{2} + 2734 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2852 T + 25156270 T^{2} + 2852 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2674 T + 58089971 T^{2} + 2674 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9146 T + 154583427 T^{2} + 9146 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6132 T + 240555334 T^{2} - 6132 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16040 T + 215636742 T^{2} + 16040 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 25326 T + 616649779 T^{2} + 25326 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14958 T + 543936427 T^{2} - 14958 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1106 T + 1400371723 T^{2} + 1106 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 28042 T + 1885112387 T^{2} + 28042 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 102642 T + 5264587579 T^{2} + 102642 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11056 T + 3430664782 T^{2} + 11056 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 35070 T + 3668746195 T^{2} - 35070 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 101762 T + 5366098883 T^{2} - 101762 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 44632 T + 2269443286 T^{2} + 44632 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 75474 T + 9250340803 T^{2} + 75474 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8316 T + 16824750934 T^{2} + 8316 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.30697904253129328471674950830, −13.22795554172326066838010918509, −12.14232957111651098303516138998, −11.99038800038381689648288103777, −10.95338205295658646990663382977, −10.85530850455147293185309232786, −10.00415297799863558905724871371, −9.508156372935485597949384234246, −8.676538073872004778623588553662, −8.336553150282396946959290561683, −7.60823490925122556101309049334, −6.68276612582128864169074864832, −6.02352680316181296462657507699, −5.81854966196110495523206875868, −5.12932517546481482712066014672, −3.95739418753189335279565774219, −3.43535089295698052586240522224, −2.97214347763689276751713737769, −1.71120446931453614057089243986, −1.39900939698162351986617373934,
1.39900939698162351986617373934, 1.71120446931453614057089243986, 2.97214347763689276751713737769, 3.43535089295698052586240522224, 3.95739418753189335279565774219, 5.12932517546481482712066014672, 5.81854966196110495523206875868, 6.02352680316181296462657507699, 6.68276612582128864169074864832, 7.60823490925122556101309049334, 8.336553150282396946959290561683, 8.676538073872004778623588553662, 9.508156372935485597949384234246, 10.00415297799863558905724871371, 10.85530850455147293185309232786, 10.95338205295658646990663382977, 11.99038800038381689648288103777, 12.14232957111651098303516138998, 13.22795554172326066838010918509, 13.30697904253129328471674950830
