| L(s) = 1 | + 79·9-s − 142·11-s + 76·19-s − 310·29-s − 176·31-s − 284·41-s + 658·49-s + 1.74e3·59-s + 890·61-s − 3.42e3·71-s + 2.54e3·79-s + 3.42e3·81-s + 1.77e3·89-s − 1.12e4·99-s + 4.11e3·101-s + 2.65e3·109-s + 7.69e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.17e3·169-s + ⋯ |
| L(s) = 1 | + 2.92·9-s − 3.89·11-s + 0.917·19-s − 1.98·29-s − 1.01·31-s − 1.08·41-s + 1.91·49-s + 3.85·59-s + 1.86·61-s − 5.72·71-s + 3.62·79-s + 4.70·81-s + 2.11·89-s − 11.3·99-s + 4.05·101-s + 2.33·109-s + 5.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.81·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.904896719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.904896719\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 79 T^{2} + 2812 T^{4} - 79 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 94 p T^{2} + 224739 T^{4} - 94 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 71 T + 3716 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6179 T^{2} + 17975772 T^{4} - 6179 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 9529 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 8781 T^{2} + 314630312 T^{4} + 8781 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 88 T + 8718 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 129356 T^{2} + 9238453302 T^{4} - 129356 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 142 T + 135458 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 80147 T^{2} + 11535366144 T^{4} - 80147 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 303051 T^{2} + 43614898952 T^{4} - 303051 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 308067 T^{2} + 58615963724 T^{4} - 308067 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 933643 T^{2} + 386067837744 T^{4} - 933643 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1712 T + 1445258 T^{2} + 1712 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 998434 T^{2} + 518766526467 T^{4} - 998434 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1274 T + 1391022 T^{2} - 1274 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2068664 T^{2} + 1722854975262 T^{4} - 2068664 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 888 T + 1554274 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 62464 T^{2} + 794547680382 T^{4} + 62464 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.24812399232755135755785153353, −5.70227832510578812644076185508, −5.57840711714795221871161933070, −5.53558858856836367447205503949, −5.47735272200369880913907269576, −5.03292400527753637608895904662, −4.99679236609114953644374533669, −4.67945618468807038191642010851, −4.56279733457070445615248073564, −4.18615505880524198882427708013, −4.01408167737283876288449496484, −3.85225537827942747335644552577, −3.38731764849325305973002141995, −3.36359321171268806717208166368, −3.15759520499613650040878052453, −2.64459952375939956105287643850, −2.54933027904869638515613247901, −2.11638641716461246127656516277, −1.95027329829287005412447894770, −1.79232660886390619511645844307, −1.71690714576729488530511716762, −0.956144762180761604091990959191, −0.70642516356963446410315979141, −0.49531504284083197837964002502, −0.42312241808437623050202799728,
0.42312241808437623050202799728, 0.49531504284083197837964002502, 0.70642516356963446410315979141, 0.956144762180761604091990959191, 1.71690714576729488530511716762, 1.79232660886390619511645844307, 1.95027329829287005412447894770, 2.11638641716461246127656516277, 2.54933027904869638515613247901, 2.64459952375939956105287643850, 3.15759520499613650040878052453, 3.36359321171268806717208166368, 3.38731764849325305973002141995, 3.85225537827942747335644552577, 4.01408167737283876288449496484, 4.18615505880524198882427708013, 4.56279733457070445615248073564, 4.67945618468807038191642010851, 4.99679236609114953644374533669, 5.03292400527753637608895904662, 5.47735272200369880913907269576, 5.53558858856836367447205503949, 5.57840711714795221871161933070, 5.70227832510578812644076185508, 6.24812399232755135755785153353
