| L(s) = 1 | − 16·23-s + 10·25-s + 6·49-s + 16·53-s − 24·79-s + 72·107-s − 40·109-s + 24·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.33·23-s + 2·25-s + 6/7·49-s + 2.19·53-s − 2.70·79-s + 6.96·107-s − 3.83·109-s + 2.25·113-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 + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7248569410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7248569410\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−5.93519193343658286202653019585, −5.58437022520374303102795641516, −5.47677645822631811091583762449, −5.31515468900529936014124106379, −4.91462802490384412718353213946, −4.78123586625999750233090330331, −4.73505505451342474198198189702, −4.44419813123629986718182962158, −4.25640024554332225054050450321, −3.93447790041513994082141400144, −3.76537166971372340244400138171, −3.71253151417336640592208087826, −3.63308042877973537761520038359, −3.09164769135147385461561817574, −2.95775605207415749850211660177, −2.68053844332892998366971013893, −2.45379452252262860207170576032, −2.44604351512203979385456186122, −1.94968490904991921614020522345, −1.76094968419382293562871541190, −1.67829232253079414466782810524, −1.17122320042817833095540258239, −0.884036760069159266331127140431, −0.67000876403028149922132624699, −0.11998043303085641195770618318,
0.11998043303085641195770618318, 0.67000876403028149922132624699, 0.884036760069159266331127140431, 1.17122320042817833095540258239, 1.67829232253079414466782810524, 1.76094968419382293562871541190, 1.94968490904991921614020522345, 2.44604351512203979385456186122, 2.45379452252262860207170576032, 2.68053844332892998366971013893, 2.95775605207415749850211660177, 3.09164769135147385461561817574, 3.63308042877973537761520038359, 3.71253151417336640592208087826, 3.76537166971372340244400138171, 3.93447790041513994082141400144, 4.25640024554332225054050450321, 4.44419813123629986718182962158, 4.73505505451342474198198189702, 4.78123586625999750233090330331, 4.91462802490384412718353213946, 5.31515468900529936014124106379, 5.47677645822631811091583762449, 5.58437022520374303102795641516, 5.93519193343658286202653019585
