| L(s) = 1 | − 486·3-s − 7.02e3·5-s − 5.65e4·7-s + 1.77e5·9-s + 5.78e5·11-s − 9.02e5·13-s + 3.41e6·15-s + 5.65e6·17-s + 1.91e7·19-s + 2.74e7·21-s − 1.23e7·23-s + 1.86e7·25-s − 5.73e7·27-s − 1.73e8·29-s + 6.26e7·31-s − 2.81e8·33-s + 3.96e8·35-s + 2.27e8·37-s + 4.38e8·39-s − 7.85e8·41-s + 8.12e8·43-s − 1.24e9·45-s + 7.11e8·47-s − 6.42e8·49-s − 2.74e9·51-s + 1.16e9·53-s − 4.06e9·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.00·5-s − 1.27·7-s + 9-s + 1.08·11-s − 0.673·13-s + 1.16·15-s + 0.966·17-s + 1.77·19-s + 1.46·21-s − 0.400·23-s + 0.381·25-s − 0.769·27-s − 1.57·29-s + 0.392·31-s − 1.25·33-s + 1.27·35-s + 0.540·37-s + 0.778·39-s − 1.05·41-s + 0.843·43-s − 1.00·45-s + 0.452·47-s − 0.324·49-s − 1.11·51-s + 0.382·53-s − 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 1404 p T + 1226958 p^{2} T^{2} + 1404 p^{12} T^{3} + p^{22} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 56520 T + 548077906 p T^{2} + 56520 p^{11} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 52600 p T + 603957635126 T^{2} - 52600 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 902068 T + 2800075958286 T^{2} + 902068 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5657652 T + 43615619702278 T^{2} - 5657652 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 19193112 T + 323650450963174 T^{2} - 19193112 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12368640 T + 1871690946611854 T^{2} + 12368640 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 173581452 T + 30058862497633518 T^{2} + 173581452 p^{11} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 62608392 T + 25015365536406478 T^{2} - 62608392 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 227986796 T + 9166167104462598 p T^{2} - 227986796 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 785693916 T + 632044485007790646 T^{2} + 785693916 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 812685096 T + 1620937325484163318 T^{2} - 812685096 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 711784720 T + 5067747011525956190 T^{2} - 711784720 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1164124836 T + 12993899229378123294 T^{2} - 1164124836 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 584580184 T + 3606361935635116886 T^{2} - 584580184 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4474864476 T + 86409579397357530062 T^{2} - 4474864476 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7319879064 T + \)\(13\!\cdots\!34\)\( T^{2} + 7319879064 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4063300576 T + 17090284326077205422 T^{2} - 4063300576 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 50128218348 T + \)\(11\!\cdots\!34\)\( T^{2} + 50128218348 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7404872328 T + \)\(13\!\cdots\!54\)\( T^{2} - 7404872328 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 43253676216 T + \)\(30\!\cdots\!22\)\( T^{2} - 43253676216 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 134083427724 T + \)\(10\!\cdots\!58\)\( T^{2} + 134083427724 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 63225345436 T + \)\(58\!\cdots\!30\)\( T^{2} + 63225345436 p^{11} T^{3} + p^{22} T^{4} \) |
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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
−11.56821487801215140290372405214, −11.27074908571852327137033180387, −10.23513000711164417687491846185, −10.06022846164936250099500338851, −9.350640490862122292133271308100, −9.078848996352783833828241440749, −7.77153785297607237571778864050, −7.68356632849271001146335794025, −6.85958597543053290550376468309, −6.58316819958839996832644267327, −5.56283055064951363992777685962, −5.53517082393053055152119173936, −4.46671328699429073269026684433, −3.87367074913414391042244915530, −3.41191857921158503251612939503, −2.70767624587993576861499984094, −1.42772073961648815222659058980, −1.00238109539328588735104352035, 0, 0,
1.00238109539328588735104352035, 1.42772073961648815222659058980, 2.70767624587993576861499984094, 3.41191857921158503251612939503, 3.87367074913414391042244915530, 4.46671328699429073269026684433, 5.53517082393053055152119173936, 5.56283055064951363992777685962, 6.58316819958839996832644267327, 6.85958597543053290550376468309, 7.68356632849271001146335794025, 7.77153785297607237571778864050, 9.078848996352783833828241440749, 9.350640490862122292133271308100, 10.06022846164936250099500338851, 10.23513000711164417687491846185, 11.27074908571852327137033180387, 11.56821487801215140290372405214
