| L(s) = 1 | + 2·3-s − 8·7-s + 9-s − 6·11-s − 16·21-s + 5·25-s + 4·27-s − 12·33-s − 16·37-s + 30·41-s − 12·47-s + 12·49-s − 12·53-s − 8·63-s + 2·67-s + 10·73-s + 10·75-s + 48·77-s − 4·81-s − 24·83-s − 6·99-s − 12·101-s − 6·107-s − 32·111-s − 11·121-s + 60·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3.02·7-s + 1/3·9-s − 1.80·11-s − 3.49·21-s + 25-s + 0.769·27-s − 2.08·33-s − 2.63·37-s + 4.68·41-s − 1.75·47-s + 12/7·49-s − 1.64·53-s − 1.00·63-s + 0.244·67-s + 1.17·73-s + 1.15·75-s + 5.47·77-s − 4/9·81-s − 2.63·83-s − 0.603·99-s − 1.19·101-s − 0.580·107-s − 3.03·111-s − 121-s + 5.41·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4509178630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4509178630\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 3 | $D_{4}$ | \( ( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - p T^{2} + 9 T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 633 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 162 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 30 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 101 T^{2} + 4185 T^{4} - 101 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 1653 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 13950 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 5169 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - T + 87 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 5 T + 105 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 11973 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 20478 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.41150368932250250680772871955, −6.33272196991110120972540135389, −6.03325097743574270061670030505, −5.77245418628657951137056206766, −5.43678790245032273724440981213, −5.42799649979192886927991131571, −5.28597592303877060713771939646, −4.74137803410842131042865470272, −4.64509546806035514743474672592, −4.61641554909603812503223250174, −4.08387066834078295097238470514, −3.92217155651008412186320171042, −3.74652266801868183118272949905, −3.34082151828458863412535332824, −3.25705491625013597938919224255, −2.83578311042653996308447391439, −2.82197321377527603898724064119, −2.81080279284915227828569898289, −2.76874914202589151159143031377, −1.96772242624668441046302056651, −1.96191210290417094245436604385, −1.51991019672301932102880842952, −1.03876516448623958378482270295, −0.53024619584692557849770158882, −0.14818706024205912780459585894,
0.14818706024205912780459585894, 0.53024619584692557849770158882, 1.03876516448623958378482270295, 1.51991019672301932102880842952, 1.96191210290417094245436604385, 1.96772242624668441046302056651, 2.76874914202589151159143031377, 2.81080279284915227828569898289, 2.82197321377527603898724064119, 2.83578311042653996308447391439, 3.25705491625013597938919224255, 3.34082151828458863412535332824, 3.74652266801868183118272949905, 3.92217155651008412186320171042, 4.08387066834078295097238470514, 4.61641554909603812503223250174, 4.64509546806035514743474672592, 4.74137803410842131042865470272, 5.28597592303877060713771939646, 5.42799649979192886927991131571, 5.43678790245032273724440981213, 5.77245418628657951137056206766, 6.03325097743574270061670030505, 6.33272196991110120972540135389, 6.41150368932250250680772871955
