| L(s) = 1 | − 12·3-s + 60·7-s + 90·9-s − 720·21-s + 400·23-s − 540·27-s − 84·29-s + 1.24e3·41-s + 192·43-s − 440·47-s + 1.53e3·49-s + 552·61-s + 5.40e3·63-s − 408·67-s − 4.80e3·69-s + 2.83e3·81-s − 608·83-s + 1.00e3·87-s + 312·89-s − 1.30e3·101-s + 396·103-s − 3.71e3·107-s + 1.53e3·109-s + 1.76e3·121-s − 1.49e4·123-s + 127-s − 2.30e3·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3.23·7-s + 10/3·9-s − 7.48·21-s + 3.62·23-s − 3.84·27-s − 0.537·29-s + 4.75·41-s + 0.680·43-s − 1.36·47-s + 4.47·49-s + 1.15·61-s + 10.7·63-s − 0.743·67-s − 8.37·69-s + 35/9·81-s − 0.804·83-s + 1.24·87-s + 0.371·89-s − 1.28·101-s + 0.378·103-s − 3.35·107-s + 1.34·109-s + 1.32·121-s − 10.9·123-s + 0.000698·127-s − 1.57·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.783665724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.783665724\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 30 T + 582 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 1768 T^{2} + 4028478 T^{4} - 1768 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 3632 T^{2} + 56382 p^{2} T^{4} - 3632 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12272 T^{2} + 83923998 T^{4} + 12272 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6124 T^{2} + 27666006 T^{4} + 6124 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 + 42 T + 46258 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 59692 T^{2} + 2433648678 T^{4} + 59692 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 28336 T^{2} - 2229036402 T^{4} + 28336 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 624 T + 202286 T^{2} - 624 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 96 T + 29718 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 220 T + 113150 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 501872 T^{2} + 106308375630 T^{4} + 501872 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 802616 T^{2} + 245338955070 T^{4} + 802616 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 276 T + 283502 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 204 T + 452694 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1289228 T^{2} + 671568403974 T^{4} + 1289228 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4468 p T^{2} + 327406862598 T^{4} + 4468 p^{7} T^{6} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1912684 T^{2} + 1400532789606 T^{4} + 1912684 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 304 T + 1119302 T^{2} + 304 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 1226518 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2557444 T^{2} + 3281668753542 T^{4} + 2557444 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−5.98719328025047429908176102383, −5.65519396399845680847612882670, −5.43827491634510086646329946505, −5.33319585302236176060375489339, −5.14018674264831131621861663174, −5.03228123482184692964014369161, −4.87065464549617657869257505296, −4.62257038981240518224777081391, −4.47939527386136724802934968734, −4.30210407640905406369937341824, −3.88489962695542546706920349257, −3.83530838719734617270578552252, −3.75769854435856559965919455768, −3.00582503449430771073182639532, −2.74925701141018656816824628326, −2.63952992591585533454615630021, −2.55257199913266261245984534511, −1.91544162027109417728830259786, −1.69266922563405071814242145542, −1.52777562574773725369275742123, −1.31474058157328371800382440188, −0.938470453527601962044448436991, −0.833470461365661071672872547127, −0.74036475558270742352683752324, −0.16712655584304290244097772353,
0.16712655584304290244097772353, 0.74036475558270742352683752324, 0.833470461365661071672872547127, 0.938470453527601962044448436991, 1.31474058157328371800382440188, 1.52777562574773725369275742123, 1.69266922563405071814242145542, 1.91544162027109417728830259786, 2.55257199913266261245984534511, 2.63952992591585533454615630021, 2.74925701141018656816824628326, 3.00582503449430771073182639532, 3.75769854435856559965919455768, 3.83530838719734617270578552252, 3.88489962695542546706920349257, 4.30210407640905406369937341824, 4.47939527386136724802934968734, 4.62257038981240518224777081391, 4.87065464549617657869257505296, 5.03228123482184692964014369161, 5.14018674264831131621861663174, 5.33319585302236176060375489339, 5.43827491634510086646329946505, 5.65519396399845680847612882670, 5.98719328025047429908176102383
