L(s) = 1 | + 2·9-s − 24·19-s − 30·25-s − 136·31-s + 68·49-s − 296·61-s + 312·79-s − 77·81-s − 296·109-s + 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 388·169-s − 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 2/9·9-s − 1.26·19-s − 6/5·25-s − 4.38·31-s + 1.38·49-s − 4.85·61-s + 3.94·79-s − 0.950·81-s − 2.71·109-s + 2.67·121-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 + 2.29·169-s − 0.280·171-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3307382738\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3307382738\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 p T^{2} + p^{4} T^{4} \) |
good | 7 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 402 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 1038 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 3042 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4398 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2718 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 514 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 7522 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15522 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 17794 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−7.07252229252093714449022158686, −6.93492592047820895183909839816, −6.43913582198432812563487540614, −6.21869073284153128396092482381, −6.05642587410979766359741297600, −5.95104285922947198801673972190, −5.60997791787082784396805071162, −5.37000575542671209361176081780, −5.19307022530198434198306701006, −4.91065164978837448556353049093, −4.62348672661812909043474462784, −4.38050865857605679457396053425, −4.00398209765649515506036594233, −3.92233544001999396905447563872, −3.68376029292752507421317710368, −3.37059281104143024782426337900, −3.19218917468931507695500974511, −2.59798153272257526442868259051, −2.55121825752279001289322625441, −1.99444385430759652777462243598, −1.76549268082112390424949835283, −1.71328367436028077459072079993, −1.21823706284386630488458937993, −0.53798223621897566175362019500, −0.11105871862258896519504784525,
0.11105871862258896519504784525, 0.53798223621897566175362019500, 1.21823706284386630488458937993, 1.71328367436028077459072079993, 1.76549268082112390424949835283, 1.99444385430759652777462243598, 2.55121825752279001289322625441, 2.59798153272257526442868259051, 3.19218917468931507695500974511, 3.37059281104143024782426337900, 3.68376029292752507421317710368, 3.92233544001999396905447563872, 4.00398209765649515506036594233, 4.38050865857605679457396053425, 4.62348672661812909043474462784, 4.91065164978837448556353049093, 5.19307022530198434198306701006, 5.37000575542671209361176081780, 5.60997791787082784396805071162, 5.95104285922947198801673972190, 6.05642587410979766359741297600, 6.21869073284153128396092482381, 6.43913582198432812563487540614, 6.93492592047820895183909839816, 7.07252229252093714449022158686