| L(s) = 1 | + 4·7-s − 4·17-s − 20·23-s + 12·25-s − 8·31-s + 4·41-s + 10·49-s + 36·71-s + 48·73-s + 8·79-s − 20·89-s − 32·97-s − 24·103-s − 16·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 80·161-s + 163-s + 167-s + 20·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.970·17-s − 4.17·23-s + 12/5·25-s − 1.43·31-s + 0.624·41-s + 10/7·49-s + 4.27·71-s + 5.61·73-s + 0.900·79-s − 2.11·89-s − 3.24·97-s − 2.36·103-s − 1.46·119-s + 1.81·121-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 + 0.0798·157-s − 6.30·161-s + 0.0783·163-s + 0.0773·167-s + 1.53·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.510358434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510358434\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 18 T + 220 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 36086 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 10 T + 200 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 246 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.66313779823959812153102196795, −6.18551449353393857299555901561, −6.09126660305747361075555799767, −5.86044081453164749999837476206, −5.69380466120448346493350099845, −5.34390676962879474448260994216, −5.22491547959676074008932550648, −5.04836046772993602730157168127, −4.91060950095146864604474394926, −4.44726052038454662811356033156, −4.37175803587553926034855918300, −4.11227506876771618846217451053, −3.96994237637406094667560154953, −3.79926149014590327908591312577, −3.38787570915286360475744204399, −3.23280942208390559914569181318, −2.96327571332404439739415450895, −2.24444579070701132031741360732, −2.16871039585241816417752557442, −2.16780332767824395728969106360, −2.11304742865446318557385712407, −1.53277523534804348153649611527, −1.05996545480053106917147034807, −0.846634010086300983164840959624, −0.28581014538454945046375747238,
0.28581014538454945046375747238, 0.846634010086300983164840959624, 1.05996545480053106917147034807, 1.53277523534804348153649611527, 2.11304742865446318557385712407, 2.16780332767824395728969106360, 2.16871039585241816417752557442, 2.24444579070701132031741360732, 2.96327571332404439739415450895, 3.23280942208390559914569181318, 3.38787570915286360475744204399, 3.79926149014590327908591312577, 3.96994237637406094667560154953, 4.11227506876771618846217451053, 4.37175803587553926034855918300, 4.44726052038454662811356033156, 4.91060950095146864604474394926, 5.04836046772993602730157168127, 5.22491547959676074008932550648, 5.34390676962879474448260994216, 5.69380466120448346493350099845, 5.86044081453164749999837476206, 6.09126660305747361075555799767, 6.18551449353393857299555901561, 6.66313779823959812153102196795
