| L(s) = 1 | − 652·4-s − 3.21e4·7-s + 1.21e6·13-s − 6.23e5·16-s − 4.20e6·19-s + 1.88e7·25-s + 2.09e7·28-s − 5.20e7·31-s + 1.03e8·37-s − 1.43e8·43-s + 2.12e8·49-s − 7.93e8·52-s − 7.13e8·61-s + 1.09e9·64-s − 1.18e9·67-s + 8.26e8·73-s + 2.74e9·76-s − 2.16e9·79-s − 3.91e10·91-s − 2.73e10·97-s − 1.22e10·100-s + 5.43e9·103-s + 1.15e10·109-s + 2.00e10·112-s − 3.42e10·121-s + 3.39e10·124-s + 127-s + ⋯ |
| L(s) = 1 | − 0.636·4-s − 1.91·7-s + 3.27·13-s − 0.594·16-s − 1.69·19-s + 1.92·25-s + 1.21·28-s − 1.81·31-s + 1.49·37-s − 0.979·43-s + 0.750·49-s − 2.08·52-s − 0.844·61-s + 1.01·64-s − 0.876·67-s + 0.398·73-s + 1.08·76-s − 0.702·79-s − 6.27·91-s − 3.18·97-s − 1.22·100-s + 0.468·103-s + 0.751·109-s + 1.13·112-s − 1.32·121-s + 1.15·124-s + 3.25·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.001614175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.001614175\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 163 p^{2} T^{2} + p^{20} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 753602 p^{2} T^{2} + p^{20} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2299 p T + p^{10} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 34254891598 T^{2} + p^{20} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 608639 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4031682981698 T^{2} + p^{20} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2104549 T + p^{10} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 61845264014498 T^{2} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 148814205399598 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 26014126 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 51946607 T + p^{10} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3226778021758 p^{2} T^{2} + p^{20} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 71964982 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82763304603064898 T^{2} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 338704040537029298 T^{2} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 654592036989174002 T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 356794201 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 591548989 T + p^{10} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 3038145413862016798 T^{2} + p^{20} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 413274143 T + p^{10} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1080519949 T + p^{10} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1472662493907013102 T^{2} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 54253153184643612002 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 13694177593 T + p^{10} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.38367696133271693151401587718, −14.76083784717747559998830546104, −13.83695938706574169493621139236, −13.17103911561533235658973373734, −13.05864674515609539914391115609, −12.57135785592600250453600463238, −11.05371226349377672254946017555, −11.03947574069782331058811435860, −10.11165857670310535883587906854, −9.071773144038974772214526069305, −8.950153022291172208419023297991, −8.220714668986603846343471400457, −6.68191520702908367163871052964, −6.47569785566710265761285746426, −5.72633782057000169233597814767, −4.33003487705448621406531219203, −3.67282368948931824460299326737, −2.97536432627674954215534665319, −1.46641241230849457656742093754, −0.40644174248457276090701261146,
0.40644174248457276090701261146, 1.46641241230849457656742093754, 2.97536432627674954215534665319, 3.67282368948931824460299326737, 4.33003487705448621406531219203, 5.72633782057000169233597814767, 6.47569785566710265761285746426, 6.68191520702908367163871052964, 8.220714668986603846343471400457, 8.950153022291172208419023297991, 9.071773144038974772214526069305, 10.11165857670310535883587906854, 11.03947574069782331058811435860, 11.05371226349377672254946017555, 12.57135785592600250453600463238, 13.05864674515609539914391115609, 13.17103911561533235658973373734, 13.83695938706574169493621139236, 14.76083784717747559998830546104, 15.38367696133271693151401587718
