| L(s) = 1 | − 16·11-s − 2·16-s + 32·19-s + 16·59-s − 64·71-s − 2·81-s + 96·89-s + 104·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 32·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 512·209-s + 211-s + ⋯ |
| L(s) = 1 | − 4.82·11-s − 1/2·16-s + 7.34·19-s + 2.08·59-s − 7.59·71-s − 2/9·81-s + 10.1·89-s + 9.45·121-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 + 0.0760·173-s + 2.41·176-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 − 35.4·209-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.305120529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.305120529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| good | 11 | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 + 142 T^{4} - 7149 T^{8} + 142 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 497 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 967 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 26 T^{2} + 1899 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2062 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 126 T^{2} + 7139 T^{4} - 126 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 5182 T^{4} + 12352611 T^{8} + 5182 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 - 4732 T^{4} + 13117830 T^{8} - 4732 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 56 T^{2} + 327 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )( 1 + 56 T^{2} + 327 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 59 | \( ( 1 - 4 T + 95 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 218 T^{2} + 19275 T^{4} - 218 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 3932 T^{4} + 41402598 T^{8} - 3932 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 16 T + 194 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 15548 T^{4} + 109241286 T^{8} - 15548 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 92 T^{2} + 12870 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 15358 T^{4} + 117552483 T^{8} + 15358 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 12 T + p T^{2} )^{8} \) |
| 97 | \( 1 + 4996 T^{4} + 20789766 T^{8} + 4996 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.17014193011251008186650316218, −4.14620063227105114682811890299, −4.10451309398996819841073958837, −4.09986449056616015612510485521, −3.58344312194057918639929999463, −3.28593552900141731424822903845, −3.17906939707581314821619776730, −3.12246173609424849319508328089, −3.10968437652499384602632168675, −3.08601665108126292115548836070, −3.06174249689958978124355591682, −2.99894127911025933011368557703, −2.80244749327818847353757221292, −2.37299797815533774382579524413, −2.34112589871768345213102526007, −2.12184274375565639226211296911, −2.00866530645534027626536649163, −1.84803002809402101133924987053, −1.67192378496122452958921133837, −1.47329313818792461910620134636, −0.998771933116725991395362808087, −0.912830866149342681982205900863, −0.825757357760506024199363831059, −0.50517866845657042615610238048, −0.34774485663229778599519760073,
0.34774485663229778599519760073, 0.50517866845657042615610238048, 0.825757357760506024199363831059, 0.912830866149342681982205900863, 0.998771933116725991395362808087, 1.47329313818792461910620134636, 1.67192378496122452958921133837, 1.84803002809402101133924987053, 2.00866530645534027626536649163, 2.12184274375565639226211296911, 2.34112589871768345213102526007, 2.37299797815533774382579524413, 2.80244749327818847353757221292, 2.99894127911025933011368557703, 3.06174249689958978124355591682, 3.08601665108126292115548836070, 3.10968437652499384602632168675, 3.12246173609424849319508328089, 3.17906939707581314821619776730, 3.28593552900141731424822903845, 3.58344312194057918639929999463, 4.09986449056616015612510485521, 4.10451309398996819841073958837, 4.14620063227105114682811890299, 4.17014193011251008186650316218
Plot not available for L-functions of degree greater than 10.
