| L(s) = 1 | − 64·2-s + 3.07e3·4-s − 1.31e5·8-s − 1.43e5·9-s + 1.19e5·11-s + 5.24e6·16-s + 9.15e6·18-s − 7.61e6·22-s + 1.24e7·23-s − 9.44e7·25-s − 2.22e7·29-s − 2.01e8·32-s − 4.39e8·36-s + 5.38e8·37-s + 2.43e9·43-s + 3.65e8·44-s − 7.99e8·46-s + 6.04e9·50-s − 2.51e9·53-s + 1.42e9·58-s + 7.51e9·64-s − 2.58e10·67-s + 1.11e10·71-s + 1.87e10·72-s − 3.44e10·74-s − 6.91e10·79-s − 1.09e10·81-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 0.807·9-s + 0.222·11-s + 5/4·16-s + 1.14·18-s − 0.315·22-s + 0.404·23-s − 1.93·25-s − 0.201·29-s − 1.06·32-s − 1.21·36-s + 1.27·37-s + 2.52·43-s + 0.334·44-s − 0.572·46-s + 2.73·50-s − 0.826·53-s + 0.285·58-s + 7/8·64-s − 2.33·67-s + 0.733·71-s + 1.14·72-s − 1.80·74-s − 2.52·79-s − 0.347·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p^{5} T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 1766 p^{4} T^{2} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3778458 p^{2} T^{2} + p^{22} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 59500 T + p^{11} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 3497402592442 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 67709469953698 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 232577084470 p^{2} T^{2} + p^{22} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6244392 T + p^{11} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 11144006 T + p^{11} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26181038059665410 T^{2} + p^{22} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 269131458 T + p^{11} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 300727880855856590 T^{2} + p^{22} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1216837084 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4390323535281893214 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 1258358642 T + p^{11} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 28795407699748377750 T^{2} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 16210090650492235354 T^{2} + p^{22} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12913106900 T + p^{11} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5573384656 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + \)\(40\!\cdots\!26\)\( T^{2} + p^{22} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 34584092840 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + \)\(11\!\cdots\!26\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + \)\(52\!\cdots\!78\)\( T^{2} + p^{22} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + \)\(87\!\cdots\!06\)\( T^{2} + p^{22} T^{4} \) |
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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
−11.09741025796439781488225684850, −11.08360396867196635838213962939, −10.27200946263806871593465076743, −9.743830894091800484675379059861, −9.105958457779297727787625573006, −9.032164922314828669898090465061, −7.972353652782064787079535092507, −7.920052881400548730428787067801, −7.22705585554209799040511840485, −6.50763184078668061140974849324, −5.83771658867962791619076019648, −5.59880703877721016604608591884, −4.38897688456522585607718221317, −3.78810020640479203294198163228, −2.74558318095411155440473210282, −2.53512847209081411497568071434, −1.53348690143844650623321750059, −1.08585024738076744901337900200, 0, 0,
1.08585024738076744901337900200, 1.53348690143844650623321750059, 2.53512847209081411497568071434, 2.74558318095411155440473210282, 3.78810020640479203294198163228, 4.38897688456522585607718221317, 5.59880703877721016604608591884, 5.83771658867962791619076019648, 6.50763184078668061140974849324, 7.22705585554209799040511840485, 7.920052881400548730428787067801, 7.972353652782064787079535092507, 9.032164922314828669898090465061, 9.105958457779297727787625573006, 9.743830894091800484675379059861, 10.27200946263806871593465076743, 11.08360396867196635838213962939, 11.09741025796439781488225684850
