| L(s) = 1 | − 4·2-s + 8·4-s − 12·8-s + 18·16-s − 16·17-s + 24·23-s + 16·31-s − 36·32-s + 64·34-s − 96·46-s − 32·47-s + 32·53-s − 16·61-s − 64·62-s + 72·64-s − 128·68-s + 64·83-s + 192·92-s + 128·94-s − 128·106-s − 8·107-s + 16·113-s + 64·121-s + 64·122-s + 128·124-s + 127-s − 132·128-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s − 4.24·8-s + 9/2·16-s − 3.88·17-s + 5.00·23-s + 2.87·31-s − 6.36·32-s + 10.9·34-s − 14.1·46-s − 4.66·47-s + 4.39·53-s − 2.04·61-s − 8.12·62-s + 9·64-s − 15.5·68-s + 7.02·83-s + 20.0·92-s + 13.2·94-s − 12.4·106-s − 0.773·107-s + 1.50·113-s + 5.81·121-s + 5.79·122-s + 11.4·124-s + 0.0887·127-s − 11.6·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.393675913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.393675913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + T^{4} )^{2} \) |
| good | 2 | \( ( 1 + p T + p T^{2} + p T^{3} + T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( 1 - 92 T^{4} - 17562 T^{8} - 92 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 12 T + 72 T^{2} - 204 T^{3} + 542 T^{4} - 204 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 64 T^{2} + 2514 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 + 292 T^{4} - 2730714 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 100 T^{2} + 5094 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 1202 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 16 T + 128 T^{2} + 880 T^{3} + 5986 T^{4} + 880 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 28 T^{2} + 4086 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 - 2972 T^{4} + 21854310 T^{8} - 2972 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 160 T^{2} + 14754 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 12868 T^{4} + 85213638 T^{8} + 12868 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 32 T + 512 T^{2} - 5984 T^{3} + 59122 T^{4} - 5984 p T^{5} + 512 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 3836 T^{4} - 51582714 T^{8} - 3836 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−3.89900989319116798220390656547, −3.72213846859977360108511642759, −3.63462562363320829961359492673, −3.60286795025651801031490287422, −3.49811738868072556831675113131, −3.36133698488011421064372335887, −2.92869610838825802508699958635, −2.92306418080405153012668301245, −2.81118344612023631643728108482, −2.79530257595619480163282864658, −2.74877814542824368579971639039, −2.66081313543374789112266804694, −2.36632382563614786072983961433, −1.98876225556966800831592543864, −1.92235166855470830278142559391, −1.89349047819643094867716794482, −1.78040914435353366617880860516, −1.74591354667854365758835895559, −1.46643015376985832702736508309, −1.06192791760985404331031506463, −0.78823675155185782254801107471, −0.77556165323936913298098558578, −0.77512274773794949832601444266, −0.46498888877697089979319011004, −0.29629875701658051895397335808,
0.29629875701658051895397335808, 0.46498888877697089979319011004, 0.77512274773794949832601444266, 0.77556165323936913298098558578, 0.78823675155185782254801107471, 1.06192791760985404331031506463, 1.46643015376985832702736508309, 1.74591354667854365758835895559, 1.78040914435353366617880860516, 1.89349047819643094867716794482, 1.92235166855470830278142559391, 1.98876225556966800831592543864, 2.36632382563614786072983961433, 2.66081313543374789112266804694, 2.74877814542824368579971639039, 2.79530257595619480163282864658, 2.81118344612023631643728108482, 2.92306418080405153012668301245, 2.92869610838825802508699958635, 3.36133698488011421064372335887, 3.49811738868072556831675113131, 3.60286795025651801031490287422, 3.63462562363320829961359492673, 3.72213846859977360108511642759, 3.89900989319116798220390656547
Plot not available for L-functions of degree greater than 10.
