| L(s) = 1 | + 6·2-s + 86·3-s − 804·4-s + 516·6-s + 4.80e3·7-s − 6.79e3·8-s − 1.04e4·9-s + 3.53e4·11-s − 6.91e4·12-s + 2.65e4·13-s + 2.88e4·14-s + 3.90e5·16-s + 4.63e5·17-s − 6.27e4·18-s − 9.25e5·19-s + 4.12e5·21-s + 2.11e5·22-s − 7.78e5·23-s − 5.84e5·24-s + 1.59e5·26-s − 7.43e5·27-s − 3.86e6·28-s − 1.00e7·29-s + 2.46e6·31-s + 4.00e6·32-s + 3.03e6·33-s + 2.78e6·34-s + ⋯ |
| L(s) = 1 | + 0.265·2-s + 0.612·3-s − 1.57·4-s + 0.162·6-s + 0.755·7-s − 0.586·8-s − 0.531·9-s + 0.727·11-s − 0.962·12-s + 0.257·13-s + 0.200·14-s + 1.49·16-s + 1.34·17-s − 0.140·18-s − 1.62·19-s + 0.463·21-s + 0.192·22-s − 0.579·23-s − 0.359·24-s + 0.0683·26-s − 0.269·27-s − 1.18·28-s − 2.62·29-s + 0.479·31-s + 0.675·32-s + 0.445·33-s + 0.357·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.870975765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870975765\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 3 p T + 105 p^{3} T^{2} - 3 p^{10} T^{3} + p^{18} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 86 T + 5954 p T^{2} - 86 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 35316 T + 2892681078 T^{2} - 35316 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 26530 T - 1541163822 T^{2} - 26530 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 463920 T + 273833245726 T^{2} - 463920 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 925426 T + 858791487510 T^{2} + 925426 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 778128 T + 3473691840430 T^{2} + 778128 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10003584 T + 52302031706070 T^{2} + 10003584 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2467260 T + 49371575832542 T^{2} - 2467260 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 30735552 T + 484209298874630 T^{2} + 30735552 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 19103448 T + 602984827739166 T^{2} + 19103448 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4065100 T + 797231337676374 T^{2} + 4065100 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 82195020 T + 3721245520696702 T^{2} - 82195020 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 55189812 T + 3123778356606670 T^{2} - 55189812 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7069218 T + 16866494212382134 T^{2} + 7069218 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 44316386 T + 21654336818123658 T^{2} - 44316386 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 241921336 T + 59516583718815510 T^{2} - 241921336 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 206493816 T + 58491352612128526 T^{2} - 206493816 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 499153188 T + 178474458263805254 T^{2} - 499153188 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5930824 p T + 239633073722978334 T^{2} - 5930824 p^{10} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 444023958 T + 333438010641681622 T^{2} + 444023958 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 636267396 T + 801539802340191990 T^{2} - 636267396 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1632716064 T + 2180562419544849758 T^{2} - 1632716064 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−11.24736840267355414168373815103, −10.73513528579450487671986428164, −10.19767358347092717573159112406, −9.681064071062166126534867535311, −9.053738390617711546116856195206, −8.736858325823682805074417413058, −8.467463836465438712973521192822, −7.920675264502913579887568848730, −7.34633947525959988117123588851, −6.57245112476261471319148351636, −5.81019747511315724849956059286, −5.32453252970888275250377022748, −5.00044365436248585046422647362, −4.08115409205300848253758962368, −3.60850188643369384816150837197, −3.59541895133901368702273437857, −2.24912981254981469504134128471, −1.86943577280585272761878224638, −0.954087512157474385978369047606, −0.34002778790331420961708499884,
0.34002778790331420961708499884, 0.954087512157474385978369047606, 1.86943577280585272761878224638, 2.24912981254981469504134128471, 3.59541895133901368702273437857, 3.60850188643369384816150837197, 4.08115409205300848253758962368, 5.00044365436248585046422647362, 5.32453252970888275250377022748, 5.81019747511315724849956059286, 6.57245112476261471319148351636, 7.34633947525959988117123588851, 7.920675264502913579887568848730, 8.467463836465438712973521192822, 8.736858325823682805074417413058, 9.053738390617711546116856195206, 9.681064071062166126534867535311, 10.19767358347092717573159112406, 10.73513528579450487671986428164, 11.24736840267355414168373815103
