| L(s) = 1 | − 2·3-s − 4·5-s + 3·9-s + 8·15-s − 4·17-s − 2·19-s + 2·25-s − 4·27-s + 12·31-s − 12·45-s + 6·49-s + 8·51-s + 4·57-s − 16·59-s − 16·61-s − 4·67-s − 16·71-s − 4·75-s − 16·79-s + 5·81-s + 16·85-s − 24·93-s + 8·95-s + 12·101-s − 16·103-s + 40·107-s + 20·121-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 9-s + 2.06·15-s − 0.970·17-s − 0.458·19-s + 2/5·25-s − 0.769·27-s + 2.15·31-s − 1.78·45-s + 6/7·49-s + 1.12·51-s + 0.529·57-s − 2.08·59-s − 2.04·61-s − 0.488·67-s − 1.89·71-s − 0.461·75-s − 1.80·79-s + 5/9·81-s + 1.73·85-s − 2.48·93-s + 0.820·95-s + 1.19·101-s − 1.57·103-s + 3.86·107-s + 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2784012081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2784012081\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 186 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.49009964094624038166094604380, −9.981165310540108862159012936753, −9.602276902858558845253128331582, −8.873051395106045766750002972322, −8.622150349541844026849169680908, −8.178868347312978205721783340220, −7.57309351257899498839697613529, −7.41683259987106922561770528688, −7.01432128788786399669100099115, −6.22263554687847045965021356679, −6.15289085109206836247261027225, −5.66044936410074556300768979170, −4.57377032110579241046194191784, −4.50440258527762922987341229427, −4.46532378233860291622697015290, −3.52518044091434973502996649912, −3.10144384723055400883782942775, −2.22504849284825007047214765418, −1.30409466606744166943503749909, −0.29126463668370005812740508918,
0.29126463668370005812740508918, 1.30409466606744166943503749909, 2.22504849284825007047214765418, 3.10144384723055400883782942775, 3.52518044091434973502996649912, 4.46532378233860291622697015290, 4.50440258527762922987341229427, 4.57377032110579241046194191784, 5.66044936410074556300768979170, 6.15289085109206836247261027225, 6.22263554687847045965021356679, 7.01432128788786399669100099115, 7.41683259987106922561770528688, 7.57309351257899498839697613529, 8.178868347312978205721783340220, 8.622150349541844026849169680908, 8.873051395106045766750002972322, 9.602276902858558845253128331582, 9.981165310540108862159012936753, 10.49009964094624038166094604380
