| L(s) = 1 | + 28·9-s − 1.08e3·29-s − 1.08e3·41-s + 652·49-s − 1.00e3·61-s − 870·81-s − 3.56e3·89-s + 2.39e3·101-s − 3.41e3·109-s − 5.00e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.90e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.03·9-s − 6.91·29-s − 4.11·41-s + 1.90·49-s − 2.09·61-s − 1.19·81-s − 4.23·89-s + 2.35·101-s − 3.00·109-s − 3.75·121-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 + 2.68·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2401667991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2401667991\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 326 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2502 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2950 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )^{2}( 1 + 8 p T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 3478 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 17574 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 57058 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58870 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 129986 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 190006 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 231190 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 404998 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 64114 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 299662 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 384050 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 908638 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 81426 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 p T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 1760830 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−6.92927615535242650374301986898, −6.74130011157621801575001352603, −6.73497284007487287629271409317, −6.31859945688544738782486835655, −5.76984612327820124771060718543, −5.71873574954797315631081575722, −5.70100986867898217778300924042, −5.31227506199860961893728034562, −5.18595101925715461520212678303, −4.89245526847840738337020848588, −4.57267198290134705285973071264, −4.02417970991783168220829642503, −3.97356073715812962746514308243, −3.96405184522719603769132011787, −3.60044272933833869922958362188, −3.31055592878097786517778576309, −3.05971990108445470609390243194, −2.45081964971306873015103317937, −2.42410657128734244217791672364, −1.81875124150651599653546107399, −1.57389152440716541784844281009, −1.53460518927197358325607773591, −1.30148192195465917127077492268, −0.30754255808547337006942640358, −0.11045903100793846961740967733,
0.11045903100793846961740967733, 0.30754255808547337006942640358, 1.30148192195465917127077492268, 1.53460518927197358325607773591, 1.57389152440716541784844281009, 1.81875124150651599653546107399, 2.42410657128734244217791672364, 2.45081964971306873015103317937, 3.05971990108445470609390243194, 3.31055592878097786517778576309, 3.60044272933833869922958362188, 3.96405184522719603769132011787, 3.97356073715812962746514308243, 4.02417970991783168220829642503, 4.57267198290134705285973071264, 4.89245526847840738337020848588, 5.18595101925715461520212678303, 5.31227506199860961893728034562, 5.70100986867898217778300924042, 5.71873574954797315631081575722, 5.76984612327820124771060718543, 6.31859945688544738782486835655, 6.73497284007487287629271409317, 6.74130011157621801575001352603, 6.92927615535242650374301986898
