| L(s) = 1 | + 14·3-s − 34·5-s − 98·7-s − 278·9-s − 420·11-s + 490·13-s − 476·15-s − 1.05e3·17-s − 1.24e3·19-s − 1.37e3·21-s − 504·23-s − 2.39e3·25-s − 7.12e3·27-s + 3.90e3·29-s + 2.04e3·31-s − 5.88e3·33-s + 3.33e3·35-s + 7.48e3·37-s + 6.86e3·39-s + 7.83e3·41-s − 1.03e4·43-s + 9.45e3·45-s + 4.19e4·47-s + 7.20e3·49-s − 1.47e4·51-s − 3.28e4·53-s + 1.42e4·55-s + ⋯ |
| L(s) = 1 | + 0.898·3-s − 0.608·5-s − 0.755·7-s − 1.14·9-s − 1.04·11-s + 0.804·13-s − 0.546·15-s − 0.886·17-s − 0.791·19-s − 0.678·21-s − 0.198·23-s − 0.766·25-s − 1.88·27-s + 0.862·29-s + 0.382·31-s − 0.939·33-s + 0.459·35-s + 0.899·37-s + 0.722·39-s + 0.727·41-s − 0.852·43-s + 0.695·45-s + 2.77·47-s + 3/7·49-s − 0.795·51-s − 1.60·53-s + 0.636·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2420502645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2420502645\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 158 p T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 34 T + 142 p^{2} T^{2} + 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 420 T + 237126 T^{2} + 420 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 490 T + 656150 T^{2} - 490 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1056 T + 3106542 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1246 T + 1618778 T^{2} + 1246 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 504 T + 12920574 T^{2} + 504 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3904 T + 26647526 T^{2} - 3904 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2044 T + 33160782 T^{2} - 2044 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7488 T + 152693494 T^{2} - 7488 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7832 T + 168604142 T^{2} - 7832 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10332 T + 248230342 T^{2} + 10332 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 41972 T + 855029710 T^{2} - 41972 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 32812 T + 664801838 T^{2} + 32812 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 48398 T + 1851574618 T^{2} + 48398 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 718 T + 1677458142 T^{2} - 718 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12824 T + 1908078582 T^{2} + 12824 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 103992 T + 6302476942 T^{2} - 103992 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 54100 T + 3096449510 T^{2} + 54100 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 64568 T + 7121481950 T^{2} - 64568 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 47810 T + 8432588842 T^{2} - 47810 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17388 T - 1489722410 T^{2} + 17388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 97296 T + 17237136142 T^{2} + 97296 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.63478178506018427936318502863, −10.06031029659649579113594073095, −9.294342858854685019458983112285, −9.262870250451126872019281470269, −8.625958692216168941136364766463, −8.276836071218806798378037341099, −7.76674478449369897042364089495, −7.72616614893169521356668778559, −6.62597626682276292426752736196, −6.46859021610813778820027951592, −5.75951582626542621150559710315, −5.50911902101865049701541958350, −4.43773254779062925298312416849, −4.27766983235689681264895631053, −3.34965852225122933882398643652, −3.19081646982074998064919755345, −2.32119607809654958738102314697, −2.28211042323654509864055031667, −0.965838570595803615284219137360, −0.12399881193292019926013275639,
0.12399881193292019926013275639, 0.965838570595803615284219137360, 2.28211042323654509864055031667, 2.32119607809654958738102314697, 3.19081646982074998064919755345, 3.34965852225122933882398643652, 4.27766983235689681264895631053, 4.43773254779062925298312416849, 5.50911902101865049701541958350, 5.75951582626542621150559710315, 6.46859021610813778820027951592, 6.62597626682276292426752736196, 7.72616614893169521356668778559, 7.76674478449369897042364089495, 8.276836071218806798378037341099, 8.625958692216168941136364766463, 9.262870250451126872019281470269, 9.294342858854685019458983112285, 10.06031029659649579113594073095, 10.63478178506018427936318502863
