| L(s) = 1 | − 3·3-s − 6·5-s + 2·7-s − 3·11-s − 4·13-s + 18·15-s + 22·19-s − 6·21-s + 48·23-s − 25-s + 27·27-s − 78·29-s + 32·31-s + 9·33-s − 12·35-s + 68·37-s + 12·39-s − 21·41-s + 61·43-s + 84·47-s + 49·49-s + 18·55-s − 66·57-s + 87·59-s + 56·61-s + 24·65-s + 31·67-s + ⋯ |
| L(s) = 1 | − 3-s − 6/5·5-s + 2/7·7-s − 0.272·11-s − 0.307·13-s + 6/5·15-s + 1.15·19-s − 2/7·21-s + 2.08·23-s − 0.0399·25-s + 27-s − 2.68·29-s + 1.03·31-s + 3/11·33-s − 0.342·35-s + 1.83·37-s + 4/13·39-s − 0.512·41-s + 1.41·43-s + 1.78·47-s + 49-s + 0.327·55-s − 1.15·57-s + 1.47·59-s + 0.918·61-s + 0.369·65-s + 0.462·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.216564092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216564092\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 11 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 124 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T - 153 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 335 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 48 T + 1297 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 78 T + 2869 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 32 T + 63 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 87 T + 6004 T^{2} - 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 56 T - 585 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9110 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 65 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 38 T - 4797 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.85198484693850715862652143880, −10.63174223248898877679554641907, −9.682108141119806545591372260100, −9.618434741317176943093975983084, −8.939877917741760961263722573382, −8.572581382397303867333208672477, −7.78957143735692214085182048858, −7.72459354763509428660668275203, −7.06654235709129802639359049203, −6.95394992354790159055253742962, −5.94765360546635617755381294071, −5.71996781703664974399292567417, −5.12685021328306403979954168342, −4.85734618119400710318371752525, −3.91498812510695906147661681574, −3.85285645144225252116132568376, −2.85318976357249950557626023844, −2.37042974589427914833582943205, −1.00575017115617042022763368146, −0.57980085730715615406488364468,
0.57980085730715615406488364468, 1.00575017115617042022763368146, 2.37042974589427914833582943205, 2.85318976357249950557626023844, 3.85285645144225252116132568376, 3.91498812510695906147661681574, 4.85734618119400710318371752525, 5.12685021328306403979954168342, 5.71996781703664974399292567417, 5.94765360546635617755381294071, 6.95394992354790159055253742962, 7.06654235709129802639359049203, 7.72459354763509428660668275203, 7.78957143735692214085182048858, 8.572581382397303867333208672477, 8.939877917741760961263722573382, 9.618434741317176943093975983084, 9.682108141119806545591372260100, 10.63174223248898877679554641907, 10.85198484693850715862652143880
