| L(s) = 1 | − 2·4-s + 8·5-s − 2·9-s + 8·11-s + 3·16-s − 16·20-s + 38·25-s − 16·29-s + 4·31-s + 4·36-s + 8·41-s − 16·44-s − 16·45-s + 4·49-s + 64·55-s − 24·59-s − 8·61-s − 4·64-s + 24·80-s + 3·81-s − 8·89-s − 16·99-s − 76·100-s + 24·109-s + 32·116-s + 12·121-s − 8·124-s + ⋯ |
| L(s) = 1 | − 4-s + 3.57·5-s − 2/3·9-s + 2.41·11-s + 3/4·16-s − 3.57·20-s + 38/5·25-s − 2.97·29-s + 0.718·31-s + 2/3·36-s + 1.24·41-s − 2.41·44-s − 2.38·45-s + 4/7·49-s + 8.62·55-s − 3.12·59-s − 1.02·61-s − 1/2·64-s + 2.68·80-s + 1/3·81-s − 0.847·89-s − 1.60·99-s − 7.59·100-s + 2.29·109-s + 2.97·116-s + 1.09·121-s − 0.718·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.230601592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.230601592\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 934 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T^{2} + 1590 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2454 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 36 T^{2} + 4694 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4966 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 236 T^{2} + 25654 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.07162826588293202923037717025, −7.02100567593980816687052085728, −6.56516401401172208262173518824, −6.44879111969137081795486182505, −6.18939560252608391018206540411, −6.06542456016819203145822753348, −5.91596310820082940209739079421, −5.52648097532418251507673643851, −5.45069266986876405512148788372, −5.35065236877848828949201858798, −5.15290828821414406664221761753, −4.39188585278430122767298644010, −4.32924006100929083793666695374, −4.30221393724165280023682052796, −4.21824711704177744823626649095, −3.29145313993073109638568645818, −3.23410035679111686880220257433, −3.11181787353464740183626680143, −2.85826137747690753053241121816, −2.07963496605641049375298459077, −1.94369054643201704776657486832, −1.80061650927439254375156084474, −1.62930283802482778087227395369, −0.959594035370683049724179886956, −0.69654635379179248564905703104,
0.69654635379179248564905703104, 0.959594035370683049724179886956, 1.62930283802482778087227395369, 1.80061650927439254375156084474, 1.94369054643201704776657486832, 2.07963496605641049375298459077, 2.85826137747690753053241121816, 3.11181787353464740183626680143, 3.23410035679111686880220257433, 3.29145313993073109638568645818, 4.21824711704177744823626649095, 4.30221393724165280023682052796, 4.32924006100929083793666695374, 4.39188585278430122767298644010, 5.15290828821414406664221761753, 5.35065236877848828949201858798, 5.45069266986876405512148788372, 5.52648097532418251507673643851, 5.91596310820082940209739079421, 6.06542456016819203145822753348, 6.18939560252608391018206540411, 6.44879111969137081795486182505, 6.56516401401172208262173518824, 7.02100567593980816687052085728, 7.07162826588293202923037717025
