| L(s) = 1 | − 3·4-s + 2·5-s + 9-s + 12·11-s + 6·16-s − 2·19-s − 6·20-s + 7·25-s − 34·29-s + 6·31-s − 3·36-s + 32·41-s − 36·44-s + 2·45-s + 22·49-s + 24·55-s − 18·59-s − 18·61-s − 10·64-s − 2·71-s + 6·76-s − 8·79-s + 12·80-s − 2·81-s + 18·89-s − 4·95-s + 12·99-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s + 1/3·9-s + 3.61·11-s + 3/2·16-s − 0.458·19-s − 1.34·20-s + 7/5·25-s − 6.31·29-s + 1.07·31-s − 1/2·36-s + 4.99·41-s − 5.42·44-s + 0.298·45-s + 22/7·49-s + 3.23·55-s − 2.34·59-s − 2.30·61-s − 5/4·64-s − 0.237·71-s + 0.688·76-s − 0.900·79-s + 1.34·80-s − 2/9·81-s + 1.90·89-s − 0.410·95-s + 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 17^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672647668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672647668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T - 3 T^{2} + 24 T^{3} - 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | \( ( 1 + T^{2} )^{3} \) |
| good | 3 | \( 1 - T^{2} + p T^{4} - 38 T^{6} + p^{3} T^{8} - p^{4} T^{10} + p^{6} T^{12} \) |
| 7 | \( 1 - 22 T^{2} + 275 T^{4} - 2316 T^{6} + 275 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 13 | \( 1 - 61 T^{2} + 1723 T^{4} - 28558 T^{6} + 1723 p^{2} T^{8} - 61 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + T + 33 T^{2} + 58 T^{3} + 33 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 70 T^{2} + 2355 T^{4} - 57580 T^{6} + 2355 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 17 T + 153 T^{2} + 936 T^{3} + 153 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 3 T + 47 T^{2} - 260 T^{3} + 47 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 10 T + 93 T^{2} - 472 T^{3} + 93 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )( 1 + 10 T + 93 T^{2} + 472 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 41 | \( ( 1 - 16 T + 183 T^{2} - 1296 T^{3} + 183 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 186 T^{2} + 16391 T^{4} - 879788 T^{6} + 16391 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 25 T^{2} + 4995 T^{4} - 125830 T^{6} + 4995 p^{2} T^{8} - 25 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - p T^{2} + 7995 T^{4} - 263774 T^{6} + 7995 p^{2} T^{8} - p^{5} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 9 T + 161 T^{2} + 902 T^{3} + 161 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 9 T + 153 T^{2} + 1064 T^{3} + 153 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( 1 - 270 T^{2} + 34775 T^{4} - 2833220 T^{6} + 34775 p^{2} T^{8} - 270 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + T + 95 T^{2} + 476 T^{3} + 95 p T^{4} + p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 421 T^{2} + 75043 T^{4} - 7247278 T^{6} + 75043 p^{2} T^{8} - 421 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 4 T + 163 T^{2} + 472 T^{3} + 163 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 170 T^{2} + 855 T^{4} + 1073140 T^{6} + 855 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 9 T + 251 T^{2} - 1442 T^{3} + 251 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 493 T^{2} + 108803 T^{4} - 13645854 T^{6} + 108803 p^{2} T^{8} - 493 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.33753172193670558649994721704, −6.98670727984606341928992060994, −6.55730885033871885714749563519, −6.37357647096275061077695188600, −6.35477118651504207866264344498, −6.04710854354530212228159614333, −5.92852871120185376302129876859, −5.60140877042469304893742287704, −5.57962699285094485382833708805, −5.50271502319871732694512477673, −5.03779651211742696822438279832, −4.66520595412946840901532946177, −4.40192245312682557445476497338, −4.23346072278715261336469673194, −4.19432427576104334395428306499, −4.07353802233461188795951944979, −3.61438390408960659081220386830, −3.57712463063428713047093127538, −3.30899342325184736431873130847, −2.71559532928683093406419897610, −2.49878211766599260146951220993, −1.89720428254307445804734850852, −1.76264529331478090908620929120, −1.30711590399809800833614645236, −0.884507012432809817552582319346,
0.884507012432809817552582319346, 1.30711590399809800833614645236, 1.76264529331478090908620929120, 1.89720428254307445804734850852, 2.49878211766599260146951220993, 2.71559532928683093406419897610, 3.30899342325184736431873130847, 3.57712463063428713047093127538, 3.61438390408960659081220386830, 4.07353802233461188795951944979, 4.19432427576104334395428306499, 4.23346072278715261336469673194, 4.40192245312682557445476497338, 4.66520595412946840901532946177, 5.03779651211742696822438279832, 5.50271502319871732694512477673, 5.57962699285094485382833708805, 5.60140877042469304893742287704, 5.92852871120185376302129876859, 6.04710854354530212228159614333, 6.35477118651504207866264344498, 6.37357647096275061077695188600, 6.55730885033871885714749563519, 6.98670727984606341928992060994, 7.33753172193670558649994721704
Plot not available for L-functions of degree greater than 10.
