| L(s) = 1 | − 6·2-s + 14·4-s + 6·5-s − 36·10-s − 48·11-s − 52·13-s − 84·16-s + 30·17-s − 64·19-s + 84·20-s + 288·22-s − 60·23-s − 175·25-s + 312·26-s − 360·29-s + 140·31-s + 216·32-s − 180·34-s − 230·37-s + 384·38-s + 234·41-s − 938·43-s − 672·44-s + 360·46-s + 618·47-s + 1.05e3·50-s − 728·52-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 7/4·4-s + 0.536·5-s − 1.13·10-s − 1.31·11-s − 1.10·13-s − 1.31·16-s + 0.428·17-s − 0.772·19-s + 0.939·20-s + 2.79·22-s − 0.543·23-s − 7/5·25-s + 2.35·26-s − 2.30·29-s + 0.811·31-s + 1.19·32-s − 0.907·34-s − 1.02·37-s + 1.63·38-s + 0.891·41-s − 3.32·43-s − 2.30·44-s + 1.15·46-s + 1.91·47-s + 2.96·50-s − 1.94·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3280486492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3280486492\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 3 p T + 11 p T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 6 T + 211 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 48 T + 2470 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 p T + 3342 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 30 T + 7699 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 64 T + 3942 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 60 T + 494 p T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 360 T + 77290 T^{2} + 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 140 T + 48930 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 230 T + 98979 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 234 T + 26683 T^{2} - 234 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 938 T + 378543 T^{2} + 938 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 618 T + 210199 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 420 T + 64606 T^{2} - 420 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 282 T + 411439 T^{2} - 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 32 T + 205386 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 544 T + 535542 T^{2} - 544 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 504 T + 736126 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 764 T + 653958 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 238 T + 125439 T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 522 T + 651823 T^{2} + 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 708 T + 1163542 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 664 T + 1779618 T^{2} + 664 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.454797620452498735747298468023, −9.201469306942394824899384266169, −8.656354916581939464317960395147, −8.255240893443178631128607904750, −7.908415229351685857426441268681, −7.78191408827438268485925146769, −7.23983179163634404191904145114, −6.85153724283759710360390840084, −6.37468083428123473232078564898, −5.61261705391939498576129662142, −5.31722524763747337099803341728, −5.15649706773807242275127136286, −4.14894315093212786988719565516, −3.97084032343759934370877409845, −3.11144669706821286879265207432, −2.42981308880365942153631857327, −1.85913980337783281874636711339, −1.81117132762125842445054730865, −0.61833833769799951755914455817, −0.31136106903687004700543830908,
0.31136106903687004700543830908, 0.61833833769799951755914455817, 1.81117132762125842445054730865, 1.85913980337783281874636711339, 2.42981308880365942153631857327, 3.11144669706821286879265207432, 3.97084032343759934370877409845, 4.14894315093212786988719565516, 5.15649706773807242275127136286, 5.31722524763747337099803341728, 5.61261705391939498576129662142, 6.37468083428123473232078564898, 6.85153724283759710360390840084, 7.23983179163634404191904145114, 7.78191408827438268485925146769, 7.908415229351685857426441268681, 8.255240893443178631128607904750, 8.656354916581939464317960395147, 9.201469306942394824899384266169, 9.454797620452498735747298468023
