| L(s) = 1 | + 12·3-s + 90·9-s + 48·11-s + 120·17-s + 48·19-s − 164·25-s + 540·27-s + 576·33-s + 408·41-s + 528·43-s − 268·49-s + 1.44e3·51-s + 576·57-s + 1.87e3·59-s + 2.35e3·67-s + 968·73-s − 1.96e3·75-s + 2.83e3·81-s + 3.40e3·83-s − 3.67e3·89-s − 1.48e3·97-s + 4.32e3·99-s + 2.83e3·107-s − 3.96e3·113-s − 2.86e3·121-s + 4.89e3·123-s + 127-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 1.31·11-s + 1.71·17-s + 0.579·19-s − 1.31·25-s + 3.84·27-s + 3.03·33-s + 1.55·41-s + 1.87·43-s − 0.781·49-s + 3.95·51-s + 1.33·57-s + 4.13·59-s + 4.28·67-s + 1.55·73-s − 3.02·75-s + 35/9·81-s + 4.50·83-s − 4.37·89-s − 1.54·97-s + 4.38·99-s + 2.55·107-s − 3.29·113-s − 2.14·121-s + 3.58·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(35.87262488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.87262488\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 + 164 T^{2} + 19542 T^{4} + 164 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 268 T^{2} - 41658 T^{4} + 268 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 24 T + 2294 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 148 T^{2} + 3687126 T^{4} + 148 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 60 T + 6118 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 24 T + 9254 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 4900 T^{2} + 206522790 T^{4} - 4900 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 56516 T^{2} + 1986668214 T^{4} + 56516 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 63916 T^{2} + 2595558 p^{2} T^{4} + 63916 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 95476 T^{2} + 7028163510 T^{4} + 95476 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 204 T + 806 p T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 264 T + 171830 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 112892 T^{2} + 23215757766 T^{4} + 112892 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 507620 T^{2} + 107623720470 T^{4} + 507620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 936 T + 498710 T^{2} - 936 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132332 T^{2} + 107394800406 T^{4} - 132332 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 1176 T + 873542 T^{2} - 1176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 4612 p T^{2} + 275267100390 T^{4} + 4612 p^{7} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 484 T + 541686 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 932332 T^{2} + 435081520806 T^{4} + 932332 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 1704 T + 1721510 T^{2} - 1704 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 1836 T + 2086774 T^{2} + 1836 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 740 T + 1943814 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.19834467779593724002973766721, −6.71455156282706224704983187238, −6.60826466676024267853482679840, −6.50198869872258064680198233561, −6.29926471320078918434562572547, −5.54171696927183071427609228332, −5.51333467420216234290244982539, −5.40638440282951443442776051888, −5.35020765071631732588429879548, −4.71310036366406086060339618376, −4.31629397416904026274936777323, −4.18080781827021538229353021800, −3.94553416345458711921712888889, −3.76330207536323289235158266297, −3.53615941841468556192844675757, −3.36084091836878453283283443734, −3.02062735189731396337932456251, −2.49382771339511400727310137239, −2.34216783460894381031231013229, −2.31803565091436193069438996837, −1.83085104661378833367304564244, −1.36937942261394892228506222379, −1.11997202332797294676443177853, −0.67657978050211941420765957109, −0.66649278665721321426528852998,
0.66649278665721321426528852998, 0.67657978050211941420765957109, 1.11997202332797294676443177853, 1.36937942261394892228506222379, 1.83085104661378833367304564244, 2.31803565091436193069438996837, 2.34216783460894381031231013229, 2.49382771339511400727310137239, 3.02062735189731396337932456251, 3.36084091836878453283283443734, 3.53615941841468556192844675757, 3.76330207536323289235158266297, 3.94553416345458711921712888889, 4.18080781827021538229353021800, 4.31629397416904026274936777323, 4.71310036366406086060339618376, 5.35020765071631732588429879548, 5.40638440282951443442776051888, 5.51333467420216234290244982539, 5.54171696927183071427609228332, 6.29926471320078918434562572547, 6.50198869872258064680198233561, 6.60826466676024267853482679840, 6.71455156282706224704983187238, 7.19834467779593724002973766721
