| L(s) = 1 | + 2·3-s + 4-s + 2·7-s + 9-s + 8·11-s + 2·12-s + 6·13-s + 18·17-s + 4·21-s + 25-s − 2·27-s + 2·28-s + 16·33-s + 36-s − 2·37-s + 12·39-s − 6·41-s + 8·44-s + 12·49-s + 36·51-s + 6·52-s + 6·53-s − 12·59-s − 6·61-s + 2·63-s − 64-s − 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 2.41·11-s + 0.577·12-s + 1.66·13-s + 4.36·17-s + 0.872·21-s + 1/5·25-s − 0.384·27-s + 0.377·28-s + 2.78·33-s + 1/6·36-s − 0.328·37-s + 1.92·39-s − 0.937·41-s + 1.20·44-s + 12/7·49-s + 5.04·51-s + 0.832·52-s + 0.824·53-s − 1.56·59-s − 0.768·61-s + 0.251·63-s − 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.52093654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.52093654\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 4 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 18 T + 165 T^{2} - 1026 T^{3} + 4796 T^{4} - 1026 p T^{5} + 165 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 28 T^{2} - 108 T^{3} + 939 T^{4} - 108 p T^{5} - 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 52 T^{2} + 108 T^{3} + 3027 T^{4} + 108 p T^{5} - 52 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 169 T^{2} + 1452 T^{3} + 13992 T^{4} + 1452 p T^{5} + 169 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T + 128 T^{2} + 696 T^{3} + 10467 T^{4} + 696 p T^{5} + 128 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 4 T - 47 T^{2} - 284 T^{3} - 1592 T^{4} - 284 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 88 T^{2} - 108 T^{3} + 7779 T^{4} - 108 p T^{5} - 88 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 24 T + 394 T^{2} + 4848 T^{3} + 49731 T^{4} + 4848 p T^{5} + 394 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T - 136 T^{2} + 52 T^{3} + 12379 T^{4} + 52 p T^{5} - 136 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 382 T^{2} - 4560 T^{3} + 45267 T^{4} - 4560 p T^{5} + 382 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 29670 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.17146187175655986042039315054, −6.69698291439977661009480293055, −6.65700620800205736714713662073, −6.53705088469656180980822695143, −6.08385233854807579923155978356, −5.84692777697462206033659513840, −5.60240420585387882821934734226, −5.59985658519175922293914364867, −5.55434903294249812600614750907, −4.97893996519387419913420726686, −4.58921834690005952464698965910, −4.37054812553548779747442319228, −4.33395475643929043177280164504, −3.71654032140345851040868495266, −3.67615239226642480937231059758, −3.48808343248662938342326992923, −3.23572598011860907327477413656, −3.16099960036251951844973149412, −2.69500991879148434362156197921, −2.42236356007080368059032455355, −1.80562580875420734450644803966, −1.54431912385779591712785244156, −1.42172925744207286899241271191, −1.13955285198319872459517780076, −0.806240403533146537900804973920,
0.806240403533146537900804973920, 1.13955285198319872459517780076, 1.42172925744207286899241271191, 1.54431912385779591712785244156, 1.80562580875420734450644803966, 2.42236356007080368059032455355, 2.69500991879148434362156197921, 3.16099960036251951844973149412, 3.23572598011860907327477413656, 3.48808343248662938342326992923, 3.67615239226642480937231059758, 3.71654032140345851040868495266, 4.33395475643929043177280164504, 4.37054812553548779747442319228, 4.58921834690005952464698965910, 4.97893996519387419913420726686, 5.55434903294249812600614750907, 5.59985658519175922293914364867, 5.60240420585387882821934734226, 5.84692777697462206033659513840, 6.08385233854807579923155978356, 6.53705088469656180980822695143, 6.65700620800205736714713662073, 6.69698291439977661009480293055, 7.17146187175655986042039315054
