| L(s) = 1 | − 16·11-s − 16·71-s − 2·81-s + 56·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 4.82·11-s − 1.89·71-s − 2/9·81-s + 5.09·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9957937124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9957937124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 11 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 13 | \( ( 1 + 146 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 254 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 734 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 3518 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 466 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 2258 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 8158 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + 17282 T^{4} + p^{4} T^{8} )^{2} \) |
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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
−3.83932049621438757278471179434, −3.74713196489849050656120661986, −3.41456694095845648937058021281, −3.34010248526607017740878040589, −3.23190168923105197902637353684, −3.20885513060841128181869111376, −3.05627344817669338494296450872, −3.04507262960608600060118068245, −2.67279158466802780000981332975, −2.59207581103366388181601252934, −2.57971626553247789576129044435, −2.55818485413719840984859380903, −2.37209702333467996906510245303, −2.20332210953874899307588174914, −1.97592587516224622648029948962, −1.94880145375329355425806551767, −1.88790313208900823872191124129, −1.52194427612685926599770805903, −1.29544757703803999667846046628, −1.26681413147620012855145036306, −0.991623075985546857984964733506, −0.898852267629923584680717301297, −0.33433051014053238099868863346, −0.29668804491520888916228260818, −0.22883945314484038727294940400,
0.22883945314484038727294940400, 0.29668804491520888916228260818, 0.33433051014053238099868863346, 0.898852267629923584680717301297, 0.991623075985546857984964733506, 1.26681413147620012855145036306, 1.29544757703803999667846046628, 1.52194427612685926599770805903, 1.88790313208900823872191124129, 1.94880145375329355425806551767, 1.97592587516224622648029948962, 2.20332210953874899307588174914, 2.37209702333467996906510245303, 2.55818485413719840984859380903, 2.57971626553247789576129044435, 2.59207581103366388181601252934, 2.67279158466802780000981332975, 3.04507262960608600060118068245, 3.05627344817669338494296450872, 3.20885513060841128181869111376, 3.23190168923105197902637353684, 3.34010248526607017740878040589, 3.41456694095845648937058021281, 3.74713196489849050656120661986, 3.83932049621438757278471179434
Plot not available for L-functions of degree greater than 10.
