| L(s) = 1 | + 128·2-s + 1.22e4·4-s + 3.12e4·5-s − 2.36e5·7-s + 1.04e6·8-s + 4.00e6·10-s − 4.01e6·11-s + 1.33e7·13-s − 3.03e7·14-s + 8.38e7·16-s + 4.76e7·17-s − 9.05e7·19-s + 3.84e8·20-s − 5.14e8·22-s + 1.02e9·23-s + 7.32e8·25-s + 1.71e9·26-s − 2.91e9·28-s − 2.07e9·29-s + 1.17e10·31-s + 6.44e9·32-s + 6.09e9·34-s − 7.40e9·35-s + 4.77e10·37-s − 1.15e10·38-s + 3.27e10·40-s − 6.78e10·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.761·7-s + 1.41·8-s + 1.26·10-s − 0.683·11-s + 0.769·13-s − 1.07·14-s + 5/4·16-s + 0.478·17-s − 0.441·19-s + 1.34·20-s − 0.967·22-s + 1.44·23-s + 3/5·25-s + 1.08·26-s − 1.14·28-s − 0.646·29-s + 2.37·31-s + 1.06·32-s + 0.676·34-s − 0.680·35-s + 3.06·37-s − 0.624·38-s + 1.26·40-s − 2.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(13.74917565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.74917565\) |
| \(L(\frac{15}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 236912 T + 17799196050 p T^{2} + 236912 p^{13} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4018560 T + 712881822242 p T^{2} + 4018560 p^{13} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 13391116 T - 38526616828530 T^{2} - 13391116 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 47615052 T + 14793777255988150 T^{2} - 47615052 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 90569648 T + 22422218733426 p T^{2} + 90569648 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1025770200 T + 1082638724302134766 T^{2} - 1025770200 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2072310228 T - 1269722572062225026 T^{2} + 2072310228 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 11728406776 T + 78923255143260763326 T^{2} - 11728406776 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 47778848716 T + \)\(10\!\cdots\!58\)\( T^{2} - 47778848716 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 67871568228 T + \)\(28\!\cdots\!38\)\( T^{2} + 67871568228 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 77614412728 T + \)\(49\!\cdots\!82\)\( T^{2} - 77614412728 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 106902703320 T + \)\(13\!\cdots\!54\)\( T^{2} + 106902703320 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4551230532 T + \)\(52\!\cdots\!02\)\( T^{2} + 4551230532 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4334346240 T + \)\(13\!\cdots\!58\)\( T^{2} - 4334346240 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 887411684860 T + \)\(51\!\cdots\!62\)\( T^{2} - 887411684860 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1467845781688 T + \)\(10\!\cdots\!10\)\( T^{2} - 1467845781688 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 936151332480 T + \)\(24\!\cdots\!22\)\( T^{2} + 936151332480 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 813936935956 T + \)\(24\!\cdots\!50\)\( T^{2} - 813936935956 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1248629414744 T + \)\(97\!\cdots\!62\)\( T^{2} + 1248629414744 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 603864311256 T - \)\(84\!\cdots\!90\)\( T^{2} + 603864311256 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8016259169484 T + \)\(56\!\cdots\!02\)\( T^{2} - 8016259169484 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1682542380284 T + \)\(74\!\cdots\!18\)\( T^{2} + 1682542380284 p^{13} T^{3} + p^{26} T^{4} \) |
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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
−11.81985002624045647758182630703, −11.31498734075581191899484002670, −10.93375245214196358365069386690, −10.13274397793240334007028213710, −9.876040369764996520922660970538, −9.259317813070838562069830868822, −8.288699237840976414601022173810, −7.989606292334089525433466826313, −6.90754570876218480283468982771, −6.67204673541375069170574623237, −5.97184931713374338844459700821, −5.71992500176022791372030333960, −4.85891664001662740416359877214, −4.55456510521502712932410666826, −3.59592234561946669702751907118, −3.11502118355381939514796724800, −2.57433305252195968040652821144, −2.05218082830931223823705038442, −1.07150956033166232483533623629, −0.69979094360226847599389386297,
0.69979094360226847599389386297, 1.07150956033166232483533623629, 2.05218082830931223823705038442, 2.57433305252195968040652821144, 3.11502118355381939514796724800, 3.59592234561946669702751907118, 4.55456510521502712932410666826, 4.85891664001662740416359877214, 5.71992500176022791372030333960, 5.97184931713374338844459700821, 6.67204673541375069170574623237, 6.90754570876218480283468982771, 7.989606292334089525433466826313, 8.288699237840976414601022173810, 9.259317813070838562069830868822, 9.876040369764996520922660970538, 10.13274397793240334007028213710, 10.93375245214196358365069386690, 11.31498734075581191899484002670, 11.81985002624045647758182630703
