| L(s) = 1 | − 1.57e3·4-s − 1.16e5·7-s − 1.52e6·13-s − 1.71e6·16-s − 2.06e7·19-s + 1.83e8·28-s + 2.12e8·31-s + 1.91e7·37-s − 3.18e9·43-s + 6.17e9·49-s + 2.40e9·52-s − 6.18e9·61-s + 9.30e9·64-s + 1.82e10·67-s − 1.24e9·73-s + 3.24e10·76-s + 2.12e10·79-s + 1.77e11·91-s − 2.63e11·97-s − 1.59e11·103-s − 5.78e10·109-s + 1.98e11·112-s − 5.44e11·121-s − 3.34e11·124-s + 127-s + 131-s + 2.39e12·133-s + ⋯ |
| L(s) = 1 | − 0.769·4-s − 2.61·7-s − 1.13·13-s − 0.407·16-s − 1.90·19-s + 2.01·28-s + 1.33·31-s + 0.0453·37-s − 3.30·43-s + 3.12·49-s + 0.876·52-s − 0.937·61-s + 1.08·64-s + 1.64·67-s − 0.0700·73-s + 1.46·76-s + 0.776·79-s + 2.97·91-s − 3.11·97-s − 1.35·103-s − 0.360·109-s + 1.06·112-s − 1.90·121-s − 1.02·124-s + 4.98·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.4055428073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4055428073\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 197 p^{3} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 8300 p T + p^{11} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 544818541222 T^{2} + p^{22} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 762650 T + p^{11} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 11975946694814 T^{2} + p^{22} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 10301704 T + p^{11} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1797531507451534 T^{2} + p^{22} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 4519838782611658 T^{2} + p^{22} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 106159508 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 9574450 T + p^{11} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1088883326741296882 T^{2} + p^{22} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1590697400 T + p^{11} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2869299998598432286 T^{2} + p^{22} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 17432708152043665114 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26931896409526885318 T^{2} + p^{22} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3092621098 T + p^{11} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 9113820400 T + p^{11} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + \)\(45\!\cdots\!42\)\( T^{2} + p^{22} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 620142950 T + p^{11} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10618486484 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(10\!\cdots\!46\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + \)\(17\!\cdots\!78\)\( T^{2} + p^{22} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 131872902350 T + p^{11} T^{2} )^{2} \) |
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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
−10.27306972014214677424072267439, −9.926753711338395529358384466252, −9.550473532620835478292094314931, −9.333957301151685202489751922628, −8.473324664934314829344872794024, −8.403810823148270814426618683137, −7.55130382473784526550574395477, −6.62101743287925088940428424289, −6.57317011872127314862130479674, −6.47601338878393177011049597221, −5.37555545339293881916220935439, −5.02932996454927522834912135273, −4.20027039529763308367304236105, −3.99978935426779498588312116118, −3.18666941895583721179564963569, −2.83279102909134229433234271406, −2.29568898286831369998046161364, −1.53092399206508955062576061958, −0.41335054186487489840419172138, −0.27377349267502761270351220858,
0.27377349267502761270351220858, 0.41335054186487489840419172138, 1.53092399206508955062576061958, 2.29568898286831369998046161364, 2.83279102909134229433234271406, 3.18666941895583721179564963569, 3.99978935426779498588312116118, 4.20027039529763308367304236105, 5.02932996454927522834912135273, 5.37555545339293881916220935439, 6.47601338878393177011049597221, 6.57317011872127314862130479674, 6.62101743287925088940428424289, 7.55130382473784526550574395477, 8.403810823148270814426618683137, 8.473324664934314829344872794024, 9.333957301151685202489751922628, 9.550473532620835478292094314931, 9.926753711338395529358384466252, 10.27306972014214677424072267439
