| L(s) = 1 | + 20·4-s + 124·16-s + 312·25-s − 816·37-s − 320·49-s + 6.24e3·100-s − 6.33e3·109-s + 1.95e4·121-s + 127-s + 131-s + 137-s + 139-s − 1.63e4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.56e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s − 6.40e3·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 5/2·4-s + 1.93·16-s + 2.49·25-s − 3.62·37-s − 0.932·49-s + 6.23·100-s − 5.56·109-s + 14.6·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 9.06·148-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.07·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s − 2.33·196-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.335727725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335727725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - 5 p T^{2} + 11 p^{3} T^{4} - 5 p^{7} T^{6} + p^{12} T^{8} )^{2} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + 160 T^{2} - 1166 p T^{4} + 160 p^{6} T^{6} + p^{12} T^{8} )^{2} \) |
| good | 5 | \( ( 1 - 78 T^{2} + 21786 T^{4} - 78 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 11 | \( ( 1 - 4882 T^{2} + 9490618 T^{4} - 4882 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 13 | \( ( 1 - 1140 T^{2} + 7965078 T^{4} - 1140 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 17 | \( ( 1 - 13534 T^{2} + 93295162 T^{4} - 13534 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 19 | \( ( 1 + 15408 T^{2} + 140280878 T^{4} + 15408 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 23 | \( ( 1 - 21826 T^{2} + 264395962 T^{4} - 21826 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 29 | \( ( 1 + 87108 T^{2} + 3078173558 T^{4} + 87108 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 31 | \( ( 1 + 55084 T^{2} + 2457753126 T^{4} + 55084 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 37 | \( ( 1 + 102 T + 96042 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} )^{8} \) |
| 41 | \( ( 1 - 19902 T^{2} + 8029011258 T^{4} - 19902 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 43 | \( ( 1 - 95856 T^{2} + 2715022542 T^{4} - 95856 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 47 | \( ( 1 - 120388 T^{2} + 511848602 p T^{4} - 120388 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 53 | \( ( 1 + 301796 T^{2} + 55951089622 T^{4} + 301796 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 59 | \( ( 1 + 4836 p T^{2} + 44136192566 T^{4} + 4836 p^{7} T^{6} + p^{12} T^{8} )^{4} \) |
| 61 | \( ( 1 - 471732 T^{2} + 42469158 p^{2} T^{4} - 471732 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 67 | \( ( 1 - 699504 T^{2} + 265609535982 T^{4} - 699504 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 71 | \( ( 1 - 409986 T^{2} + 67355541306 T^{4} - 409986 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 73 | \( ( 1 + 432412 T^{2} + 213304943014 T^{4} + 432412 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 79 | \( ( 1 - 1463328 T^{2} + 978005421438 T^{4} - 1463328 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 83 | \( ( 1 + 894380 T^{2} + 846255562198 T^{4} + 894380 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 89 | \( ( 1 - 2814606 T^{2} + 2974460726106 T^{4} - 2814606 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
| 97 | \( ( 1 - 2731524 T^{2} + 3359575902342 T^{4} - 2731524 p^{6} T^{6} + p^{12} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.93687167606777282471949190126, −2.67705866873337113186705933571, −2.63649497794506626634702703862, −2.59268138346202208652667003505, −2.58083611563614278548706164829, −2.28152200223224802437867139132, −2.22534733679483035976098136585, −2.19912664928919647487852459893, −2.18386791455983477343098432285, −1.95613382555910773866505804529, −1.79511167379903828997330764014, −1.75532572344079101547248808512, −1.70578543789584396373929584182, −1.68340038714252014356930161032, −1.43408465975848932114952003617, −1.37006523864020470581703312032, −1.27368779829303292789361852688, −1.03166803310029350256070513454, −0.967971998143645836277459062665, −0.74575324925323944111373416928, −0.71238201867175026072348015098, −0.48801721204401474761629183297, −0.43915067889361789248598756622, −0.27344003197554614384379480831, −0.04049908989939986391806504233,
0.04049908989939986391806504233, 0.27344003197554614384379480831, 0.43915067889361789248598756622, 0.48801721204401474761629183297, 0.71238201867175026072348015098, 0.74575324925323944111373416928, 0.967971998143645836277459062665, 1.03166803310029350256070513454, 1.27368779829303292789361852688, 1.37006523864020470581703312032, 1.43408465975848932114952003617, 1.68340038714252014356930161032, 1.70578543789584396373929584182, 1.75532572344079101547248808512, 1.79511167379903828997330764014, 1.95613382555910773866505804529, 2.18386791455983477343098432285, 2.19912664928919647487852459893, 2.22534733679483035976098136585, 2.28152200223224802437867139132, 2.58083611563614278548706164829, 2.59268138346202208652667003505, 2.63649497794506626634702703862, 2.67705866873337113186705933571, 2.93687167606777282471949190126
Plot not available for L-functions of degree greater than 10.
