| L(s) = 1 | + 4·5-s + 8·7-s + 4·9-s + 8·11-s + 12·13-s − 4·25-s + 32·35-s − 24·43-s + 16·45-s + 34·49-s + 32·55-s − 12·61-s + 32·63-s + 48·65-s + 24·67-s + 64·77-s + 6·81-s + 96·91-s + 32·99-s − 4·101-s + 48·103-s − 8·107-s − 24·113-s + 48·117-s + 20·121-s − 44·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 3.02·7-s + 4/3·9-s + 2.41·11-s + 3.32·13-s − 4/5·25-s + 5.40·35-s − 3.65·43-s + 2.38·45-s + 34/7·49-s + 4.31·55-s − 1.53·61-s + 4.03·63-s + 5.95·65-s + 2.93·67-s + 7.29·77-s + 2/3·81-s + 10.0·91-s + 3.21·99-s − 0.398·101-s + 4.72·103-s − 0.773·107-s − 2.25·113-s + 4.43·117-s + 1.81·121-s − 3.93·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.63525355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.63525355\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 1686 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 19530 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.32709335087679600290435756314, −6.87350587358488049959631539914, −6.75082922730444442436704367243, −6.48558041224434865201750243035, −6.41324342318245343264630003546, −5.98146381498554191637788670242, −5.89393277229525051921816937659, −5.85600286798216815119185839286, −5.46086489129919894735645032491, −5.03633897638830039365118085678, −4.94999344189708963318868152447, −4.64608611656547860784314597478, −4.58427785639446960445406731621, −4.02016285452699357988155122382, −3.86756551138795047522610459763, −3.76634206770970039153092937335, −3.50155169174521143539426026060, −3.30848256172512858456214678087, −2.45167436813985850173032325578, −2.12368546178447106337276691732, −1.91856583621132483853471710087, −1.61340284455675737489073043688, −1.41789064091893615861703309139, −1.26493397192796810508723456933, −1.07603577041752416904336029184,
1.07603577041752416904336029184, 1.26493397192796810508723456933, 1.41789064091893615861703309139, 1.61340284455675737489073043688, 1.91856583621132483853471710087, 2.12368546178447106337276691732, 2.45167436813985850173032325578, 3.30848256172512858456214678087, 3.50155169174521143539426026060, 3.76634206770970039153092937335, 3.86756551138795047522610459763, 4.02016285452699357988155122382, 4.58427785639446960445406731621, 4.64608611656547860784314597478, 4.94999344189708963318868152447, 5.03633897638830039365118085678, 5.46086489129919894735645032491, 5.85600286798216815119185839286, 5.89393277229525051921816937659, 5.98146381498554191637788670242, 6.41324342318245343264630003546, 6.48558041224434865201750243035, 6.75082922730444442436704367243, 6.87350587358488049959631539914, 7.32709335087679600290435756314
