| L(s) = 1 | − 12·19-s + 42·37-s − 11·64-s + 42·73-s + 132·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + ⋯ |
| L(s) = 1 | − 2.75·19-s + 6.90·37-s − 1.37·64-s + 4.91·73-s + 12.6·109-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.626842516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.626842516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 11 T^{6} + 57 T^{12} + 11 p^{6} T^{18} + p^{12} T^{24} \) |
| 5 | \( 1 + 182 T^{6} + 17499 T^{12} + 182 p^{6} T^{18} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 34 T^{3} + 813 T^{6} - 34 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 11 | \( 1 + 2630 T^{6} + 5145339 T^{12} + 2630 p^{6} T^{18} + p^{12} T^{24} \) |
| 13 | \( ( 1 + 38 T^{3} - 753 T^{6} + 38 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 7 T^{2} - 240 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 19 | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{6} \) |
| 23 | \( 1 + 14654 T^{6} + 66703827 T^{12} + 14654 p^{6} T^{18} + p^{12} T^{24} \) |
| 29 | \( 1 - 27610 T^{6} + 167488779 T^{12} - 27610 p^{6} T^{18} + p^{12} T^{24} \) |
| 31 | \( ( 1 - 232 T^{3} + 24033 T^{6} - 232 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{6} \) |
| 41 | \( 1 + 132158 T^{6} + 12715632723 T^{12} + 132158 p^{6} T^{18} + p^{12} T^{24} \) |
| 43 | \( ( 1 - 250 T^{3} - 17007 T^{6} - 250 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 + 207506 T^{6} + 32279524707 T^{12} + 207506 p^{6} T^{18} + p^{12} T^{24} \) |
| 53 | \( ( 1 + p T^{2} )^{12} \) |
| 59 | \( 1 - 367558 T^{6} + 92918349723 T^{12} - 367558 p^{6} T^{18} + p^{12} T^{24} \) |
| 61 | \( ( 1 + 938 T^{3} + 652863 T^{6} + 938 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 1010 T^{3} + 719337 T^{6} + 1010 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 34 T^{2} - 3885 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 73 | \( ( 1 - 17 T + p T^{2} )^{6}( 1 + 10 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 - 466 T^{3} - 275883 T^{6} - 466 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 - 519766 T^{6} - 56783678613 T^{12} - 519766 p^{6} T^{18} + p^{12} T^{24} \) |
| 89 | \( ( 1 - 151 T^{2} + 14880 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 97 | \( ( 1 - 574 T^{3} - 583197 T^{6} - 574 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.28663400858056761761374427295, −3.15915587563403282201934958528, −3.15567778360861080908107442461, −3.12923648066910479015380413594, −3.07897303255873672720151199643, −2.92119052823457715338233266095, −2.59480105021525445966175096067, −2.51496590107378293152287909613, −2.50794967097364466926721323671, −2.37577849501833945058197406608, −2.30922252866518167977126876976, −2.26050320603428414058540288375, −2.25357579427254747573971788132, −2.14280854088380972807815994625, −1.88055184759547715798395073810, −1.76268638340147469963111950755, −1.70744577144327358982726639551, −1.39018499339414167681774285656, −1.15006501494854554047007508715, −1.14808522248618584994817858270, −0.942602424970745451216896795516, −0.900752997348352219079508655158, −0.76014569404689579433107310381, −0.33661693654214458371783424770, −0.25732361612947361116452664767,
0.25732361612947361116452664767, 0.33661693654214458371783424770, 0.76014569404689579433107310381, 0.900752997348352219079508655158, 0.942602424970745451216896795516, 1.14808522248618584994817858270, 1.15006501494854554047007508715, 1.39018499339414167681774285656, 1.70744577144327358982726639551, 1.76268638340147469963111950755, 1.88055184759547715798395073810, 2.14280854088380972807815994625, 2.25357579427254747573971788132, 2.26050320603428414058540288375, 2.30922252866518167977126876976, 2.37577849501833945058197406608, 2.50794967097364466926721323671, 2.51496590107378293152287909613, 2.59480105021525445966175096067, 2.92119052823457715338233266095, 3.07897303255873672720151199643, 3.12923648066910479015380413594, 3.15567778360861080908107442461, 3.15915587563403282201934958528, 3.28663400858056761761374427295
Plot not available for L-functions of degree greater than 10.
