| L(s) = 1 | + 2-s − 5·3-s − 11·4-s + 3·5-s − 5·6-s − 15·8-s + 3·9-s + 3·10-s + 80·11-s + 55·12-s + 26·13-s − 15·15-s + 61·16-s − 19·17-s + 3·18-s + 84·19-s − 33·20-s + 80·22-s + 196·23-s + 75·24-s − 239·25-s + 26·26-s − 40·27-s − 44·29-s − 15·30-s + 86·31-s + 89·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.962·3-s − 1.37·4-s + 0.268·5-s − 0.340·6-s − 0.662·8-s + 1/9·9-s + 0.0948·10-s + 2.19·11-s + 1.32·12-s + 0.554·13-s − 0.258·15-s + 0.953·16-s − 0.271·17-s + 0.0392·18-s + 1.01·19-s − 0.368·20-s + 0.775·22-s + 1.77·23-s + 0.637·24-s − 1.91·25-s + 0.196·26-s − 0.285·27-s − 0.281·29-s − 0.0912·30-s + 0.498·31-s + 0.491·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.881924305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881924305\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 5 T + 22 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 248 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 19 T + 8688 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 84 T + 11130 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 44 T + 10094 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 86 T + 56518 T^{2} - 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 230 T + 149010 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 435 T + 192728 T^{2} + 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 368 T + 379266 T^{2} - 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1058 T + 580378 T^{2} - 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 68 T + 373930 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 456 T + 542718 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1958 T + 1961238 T^{2} + 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 720 T + 899726 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 928 T + 943870 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.13877387658825147480393978269, −9.933904802399837010005903956456, −9.521045511277581739289449612430, −9.164430755482853945450105255820, −8.649041587833508363968940044942, −8.598576075311458985079793259617, −7.53120268845669819363429628843, −7.40723318258213383083132241421, −6.51070563234251516058531632578, −6.34124834219563152893201975414, −5.61852337632196502864439090358, −5.60265507239413258775100229653, −4.83480671948450839547481616176, −4.37535940135865106321942376167, −3.89014916760756351324370513641, −3.60165530160621362986927127363, −2.75874887047896842734202149635, −1.65998742274114912024225092113, −1.02904439195166679039200908202, −0.51574512209756138484841853578,
0.51574512209756138484841853578, 1.02904439195166679039200908202, 1.65998742274114912024225092113, 2.75874887047896842734202149635, 3.60165530160621362986927127363, 3.89014916760756351324370513641, 4.37535940135865106321942376167, 4.83480671948450839547481616176, 5.60265507239413258775100229653, 5.61852337632196502864439090358, 6.34124834219563152893201975414, 6.51070563234251516058531632578, 7.40723318258213383083132241421, 7.53120268845669819363429628843, 8.598576075311458985079793259617, 8.649041587833508363968940044942, 9.164430755482853945450105255820, 9.521045511277581739289449612430, 9.933904802399837010005903956456, 10.13877387658825147480393978269
