| L(s) = 1 | + 88·13-s + 10·25-s + 8·37-s + 76·49-s − 124·61-s + 304·73-s − 128·97-s − 812·109-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.16e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 6.76·13-s + 2/5·25-s + 8/37·37-s + 1.55·49-s − 2.03·61-s + 4.16·73-s − 1.31·97-s − 7.44·109-s + 3.20·121-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 + 24.6·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.02233908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.02233908\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 533 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 13 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 191 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1547 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3458 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4226 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 5573 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 6374 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 8438 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 2206 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12107 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 3097 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 12962 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{4} \) |
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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
−6.48531317644886442151920960884, −6.07542827416156136072810739620, −5.72279026492969566119567566357, −5.59331053482706433376081585094, −5.55999800126947305266471731331, −5.51078574170880522440764660094, −5.13039237013168219602335900508, −4.56132097123238993334321946244, −4.50289393042389497749177703367, −4.15945081184647979171306654079, −4.03423783065252859800427068087, −3.92465945451352093478958912425, −3.65587893003371432876768009402, −3.34257168041660878304482728432, −3.32690525519222437504693488878, −3.11679044047596477084062137919, −2.76823742969333761932864280262, −2.25917618536470218945845711803, −2.16377751141339772657054805243, −1.59680169695039543470418274483, −1.39575439400212836757417761253, −1.25561480993411243329743980707, −1.12879341483121352417393478149, −0.56712017170961588831119869205, −0.54245034668820873482215305507,
0.54245034668820873482215305507, 0.56712017170961588831119869205, 1.12879341483121352417393478149, 1.25561480993411243329743980707, 1.39575439400212836757417761253, 1.59680169695039543470418274483, 2.16377751141339772657054805243, 2.25917618536470218945845711803, 2.76823742969333761932864280262, 3.11679044047596477084062137919, 3.32690525519222437504693488878, 3.34257168041660878304482728432, 3.65587893003371432876768009402, 3.92465945451352093478958912425, 4.03423783065252859800427068087, 4.15945081184647979171306654079, 4.50289393042389497749177703367, 4.56132097123238993334321946244, 5.13039237013168219602335900508, 5.51078574170880522440764660094, 5.55999800126947305266471731331, 5.59331053482706433376081585094, 5.72279026492969566119567566357, 6.07542827416156136072810739620, 6.48531317644886442151920960884
