| L(s) = 1 | + 24·23-s − 10·25-s − 4·49-s − 24·53-s + 16·79-s − 48·107-s − 40·109-s + 24·113-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 5.00·23-s − 2·25-s − 4/7·49-s − 3.29·53-s + 1.80·79-s − 4.64·107-s − 3.83·109-s + 2.25·113-s + 3.63·121-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 − 2.46·169-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
| \(L(1)\) |
\(\approx\) |
\(3.692591682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.692591682\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−5.79865698310958608378890611867, −5.64633420223358100477090752055, −5.27605654847016215946260205112, −5.09262789493234766328709347162, −5.05952789454195759461930693067, −5.02194831694716501588023142008, −4.56985802559239277515004276227, −4.53360190632564651329760997313, −4.36492441558904919274285654059, −4.05279114012434866426237284558, −3.76227282588229410019788834516, −3.45628457355621137016174814386, −3.37815400944779127426623158061, −3.26217684815086768875138559224, −3.07851626062521577420700557667, −2.75450030612300859736008839951, −2.47012138366206564210966438207, −2.29691893443410717270073132287, −2.22177909883396032332225653666, −1.50496017323987268878661003682, −1.47673601985993294559826549583, −1.33264899731330445332699026825, −1.06551986943002071968702390171, −0.48644593039133164950878653791, −0.32535678152412635215937205738,
0.32535678152412635215937205738, 0.48644593039133164950878653791, 1.06551986943002071968702390171, 1.33264899731330445332699026825, 1.47673601985993294559826549583, 1.50496017323987268878661003682, 2.22177909883396032332225653666, 2.29691893443410717270073132287, 2.47012138366206564210966438207, 2.75450030612300859736008839951, 3.07851626062521577420700557667, 3.26217684815086768875138559224, 3.37815400944779127426623158061, 3.45628457355621137016174814386, 3.76227282588229410019788834516, 4.05279114012434866426237284558, 4.36492441558904919274285654059, 4.53360190632564651329760997313, 4.56985802559239277515004276227, 5.02194831694716501588023142008, 5.05952789454195759461930693067, 5.09262789493234766328709347162, 5.27605654847016215946260205112, 5.64633420223358100477090752055, 5.79865698310958608378890611867
