| L(s) = 1 | + 56·13-s + 84·25-s − 240·37-s + 14·49-s + 40·61-s + 168·73-s − 504·97-s + 232·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.28e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4.30·13-s + 3.35·25-s − 6.48·37-s + 2/7·49-s + 0.655·61-s + 2.30·73-s − 5.19·97-s + 1.91·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.59·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.306172207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.306172207\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 546 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 932 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1232 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3290 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3250 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 1674 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 1568 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2426 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 4246 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 2676 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 4390 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 5714 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14490 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 126 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.35716577250080086377097017869, −6.23656620505701017135398745598, −5.72384130122422520455390727884, −5.62220650021584548563078901357, −5.58221319353650946829210680075, −5.27943857388504488591743725928, −5.04372008737311590851597259409, −5.04203533927824156488103526507, −4.48566996308456896526411324341, −4.33353829815907852181285133246, −4.17990122999406174506943009693, −3.71272992641844800596915943840, −3.61784702470571300595214299146, −3.44435697962155966640054443962, −3.27161248451363389328863489820, −3.22213322285868767786353328516, −2.72497704272433966989997059171, −2.50365708088043162814891096950, −1.89752806920385654684836710101, −1.85796536741045534152926233297, −1.39008708916359184127455444556, −1.35450073549116109720946224178, −1.06199693249631494504189405223, −0.65872557759073809367234778678, −0.29712121738179478302216977119,
0.29712121738179478302216977119, 0.65872557759073809367234778678, 1.06199693249631494504189405223, 1.35450073549116109720946224178, 1.39008708916359184127455444556, 1.85796536741045534152926233297, 1.89752806920385654684836710101, 2.50365708088043162814891096950, 2.72497704272433966989997059171, 3.22213322285868767786353328516, 3.27161248451363389328863489820, 3.44435697962155966640054443962, 3.61784702470571300595214299146, 3.71272992641844800596915943840, 4.17990122999406174506943009693, 4.33353829815907852181285133246, 4.48566996308456896526411324341, 5.04203533927824156488103526507, 5.04372008737311590851597259409, 5.27943857388504488591743725928, 5.58221319353650946829210680075, 5.62220650021584548563078901357, 5.72384130122422520455390727884, 6.23656620505701017135398745598, 6.35716577250080086377097017869
