| L(s) = 1 | − 4·7-s + 4·9-s − 8·17-s − 16·23-s + 12·25-s + 8·41-s + 32·47-s + 10·49-s − 16·63-s − 32·71-s + 24·73-s + 32·79-s + 2·81-s − 8·89-s − 40·97-s − 32·103-s + 24·113-s + 32·119-s + 12·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 32·153-s + 157-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 4/3·9-s − 1.94·17-s − 3.33·23-s + 12/5·25-s + 1.24·41-s + 4.66·47-s + 10/7·49-s − 2.01·63-s − 3.79·71-s + 2.80·73-s + 3.60·79-s + 2/9·81-s − 0.847·89-s − 4.06·97-s − 3.15·103-s + 2.25·113-s + 2.93·119-s + 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 2.58·153-s + 0.0798·157-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
| \(L(1)\) |
\(\approx\) |
\(1.007510473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007510473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 - 4 T^{2} + 14 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 12 T^{2} + 150 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 12 T^{2} - 18 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 36 T^{2} + 1038 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 44 T^{2} + 2038 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 116 T^{2} + 5974 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 140 T^{2} + 8470 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 84 T^{2} + 5334 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 196 T^{2} + 16174 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 44 T^{2} + 5614 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 204 T^{2} + 18870 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 132 T^{2} + 13134 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−7.15124248992741200794844241104, −6.88485917240661119438985936688, −6.85433464557324392681858582816, −6.60956141677591059337507567762, −6.57541740603830524177403390152, −5.91178159885015951601624762899, −5.88253989446817293065956727907, −5.76694528978135966106523776545, −5.75409691354189271580749715436, −5.02969442222668144545760792905, −4.86674656817961731328386399614, −4.58121578400067069975973221691, −4.31907129274069995006206391116, −4.10580429523943148614150629175, −3.96903067483209967287134520277, −3.64335159958302093102705607241, −3.61024147735389999189844250735, −2.77590401298189243668772969266, −2.73586866346125114337555037798, −2.54655482498280317382189571805, −2.11700464203147875685397709880, −1.96274941220331534526711784378, −1.20216301084953963803582209224, −1.00603784400427532290366559574, −0.26990162487436964217273353815,
0.26990162487436964217273353815, 1.00603784400427532290366559574, 1.20216301084953963803582209224, 1.96274941220331534526711784378, 2.11700464203147875685397709880, 2.54655482498280317382189571805, 2.73586866346125114337555037798, 2.77590401298189243668772969266, 3.61024147735389999189844250735, 3.64335159958302093102705607241, 3.96903067483209967287134520277, 4.10580429523943148614150629175, 4.31907129274069995006206391116, 4.58121578400067069975973221691, 4.86674656817961731328386399614, 5.02969442222668144545760792905, 5.75409691354189271580749715436, 5.76694528978135966106523776545, 5.88253989446817293065956727907, 5.91178159885015951601624762899, 6.57541740603830524177403390152, 6.60956141677591059337507567762, 6.85433464557324392681858582816, 6.88485917240661119438985936688, 7.15124248992741200794844241104
