| L(s) = 1 | − 104·25-s − 380·49-s − 200·73-s − 568·97-s − 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.24e3·169-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 + ⋯ |
| L(s) = 1 | − 4.15·25-s − 7.75·49-s − 2.73·73-s − 5.85·97-s − 5.61·121-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 + 7.36·169-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 + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01915792472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01915792472\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 26 T^{2} + p^{4} T^{4} )^{4} \) |
| 7 | \( ( 1 + 95 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 - 311 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 70 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 697 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 458 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 + 1730 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 1655 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4394 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 5522 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 + 1130 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 - 6935 T^{2} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8617 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 5378 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + 25 T + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 7201 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 + 3410 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 - 15770 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 71 T + p^{2} T^{2} )^{8} \) |
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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
−3.64357463623479286001161519602, −3.57687976133195778918853748999, −3.51997354301295083176078132139, −3.44534961087600860397628721281, −3.29046534992190201925510319570, −3.08552694168623057054353920629, −2.80248196907302696212327813520, −2.76840751761834773182240751640, −2.75063029059348235117818650543, −2.62938229167158823320327107448, −2.47025765612019520472340109294, −2.41112442751050100054091172338, −2.00316935279502300939463428374, −1.88030231727337441014917715727, −1.79026886765337345102507486722, −1.61894032869533742370819756708, −1.60803613586553424242282463209, −1.41034842124828372133635170184, −1.39079489620171039806435681562, −1.24853302090461583480361544561, −0.878000873276693675359906249040, −0.60290116123172740402373703382, −0.33531751018756129309494605214, −0.099191983489930710560631224315, −0.04243087360422217423076410126,
0.04243087360422217423076410126, 0.099191983489930710560631224315, 0.33531751018756129309494605214, 0.60290116123172740402373703382, 0.878000873276693675359906249040, 1.24853302090461583480361544561, 1.39079489620171039806435681562, 1.41034842124828372133635170184, 1.60803613586553424242282463209, 1.61894032869533742370819756708, 1.79026886765337345102507486722, 1.88030231727337441014917715727, 2.00316935279502300939463428374, 2.41112442751050100054091172338, 2.47025765612019520472340109294, 2.62938229167158823320327107448, 2.75063029059348235117818650543, 2.76840751761834773182240751640, 2.80248196907302696212327813520, 3.08552694168623057054353920629, 3.29046534992190201925510319570, 3.44534961087600860397628721281, 3.51997354301295083176078132139, 3.57687976133195778918853748999, 3.64357463623479286001161519602
Plot not available for L-functions of degree greater than 10.
