| L(s) = 1 | + 4·9-s + 8·11-s − 20·25-s − 8·43-s + 20·49-s + 72·67-s + 16·81-s + 32·99-s + 88·107-s − 72·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 4/3·9-s + 2.41·11-s − 4·25-s − 1.21·43-s + 20/7·49-s + 8.79·67-s + 16/9·81-s + 3.21·99-s + 8.50·107-s − 6.77·113-s + 2.90·121-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 − 2.76·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.96270251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.96270251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 2 T^{2} - 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 2 p T^{2} + 54 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 18 T^{2} + 230 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 28 T^{2} + 438 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 10 T^{2} + 558 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 48 T^{2} + 1550 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 84 T^{2} + 3350 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 - 124 T^{2} + 6870 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 148 T^{2} + 9558 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 202 T^{2} + 16974 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 90 T^{2} + 8438 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 18 T + 194 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 322 T^{2} + 39678 T^{4} - 322 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 28 T^{2} - 8202 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.79781647014352589463826707893, −3.76616274348215962857181519170, −3.75545298496936603376937291271, −3.70268970610003542880384569088, −3.67710925102367295729955137250, −3.46325434731883521040879250602, −3.16774087700123960125128754572, −2.97765366499727836311035204398, −2.97645925879295602366102554995, −2.74731856434269940526238662491, −2.44345315210792354828808247144, −2.43271635073889260157343784549, −2.33015997263061048933484774317, −2.18112729203969994249443034554, −1.98787996386026924317276219210, −1.94340613321314810864428205714, −1.73462730846083983670954231484, −1.62467976026801286933443335400, −1.58111784207210820536979757737, −1.14808382048185877774186369158, −1.08500392096782468753226132629, −0.910932006299196961649761993976, −0.72006819822398289212846816617, −0.48603493533689202500437225464, −0.31066792857516099327119075745,
0.31066792857516099327119075745, 0.48603493533689202500437225464, 0.72006819822398289212846816617, 0.910932006299196961649761993976, 1.08500392096782468753226132629, 1.14808382048185877774186369158, 1.58111784207210820536979757737, 1.62467976026801286933443335400, 1.73462730846083983670954231484, 1.94340613321314810864428205714, 1.98787996386026924317276219210, 2.18112729203969994249443034554, 2.33015997263061048933484774317, 2.43271635073889260157343784549, 2.44345315210792354828808247144, 2.74731856434269940526238662491, 2.97645925879295602366102554995, 2.97765366499727836311035204398, 3.16774087700123960125128754572, 3.46325434731883521040879250602, 3.67710925102367295729955137250, 3.70268970610003542880384569088, 3.75545298496936603376937291271, 3.76616274348215962857181519170, 3.79781647014352589463826707893
Plot not available for L-functions of degree greater than 10.
