| L(s) = 1 | + 4·3-s + 4·7-s + 7·9-s − 16·13-s + 68·19-s + 16·21-s − 8·27-s − 28·31-s − 112·37-s − 64·39-s + 16·43-s − 86·49-s + 272·57-s − 92·61-s + 28·63-s + 64·67-s + 212·73-s + 44·79-s − 95·81-s − 64·91-s − 112·93-s − 244·97-s − 92·103-s + 172·109-s − 448·111-s − 112·117-s + 62·121-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 4/7·7-s + 7/9·9-s − 1.23·13-s + 3.57·19-s + 0.761·21-s − 0.296·27-s − 0.903·31-s − 3.02·37-s − 1.64·39-s + 0.372·43-s − 1.75·49-s + 4.77·57-s − 1.50·61-s + 4/9·63-s + 0.955·67-s + 2.90·73-s + 0.556·79-s − 1.17·81-s − 0.703·91-s − 1.20·93-s − 2.51·97-s − 0.893·103-s + 1.57·109-s − 4.03·111-s − 0.957·117-s + 0.512·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.303779341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.303779341\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 4 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 398 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2642 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2798 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3998 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6782 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 32 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7202 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 106 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 802 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 122 T + p^{2} T^{2} )^{2} \) |
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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
−9.832189284610846138830258949718, −9.311510540105427808001387077179, −9.141427446431982795287675525591, −8.317217060703710671569510490230, −8.279918374656114673852381301808, −7.59608287421414082847734504070, −7.51701317787040087978951613833, −7.05966473800081490091249911134, −6.72003899993339773950493315246, −5.85318041918418322686181363906, −5.23400535923897043543176589766, −5.17336297482945560086563380641, −4.78063454465715756125942947362, −3.73190479375776522173487552390, −3.63307989257964635870023343004, −2.99683592094303741151212722225, −2.71893974214389034704647807866, −1.74103949034867495749607400494, −1.63264973282916928441493403462, −0.54881460796472488261253561709,
0.54881460796472488261253561709, 1.63264973282916928441493403462, 1.74103949034867495749607400494, 2.71893974214389034704647807866, 2.99683592094303741151212722225, 3.63307989257964635870023343004, 3.73190479375776522173487552390, 4.78063454465715756125942947362, 5.17336297482945560086563380641, 5.23400535923897043543176589766, 5.85318041918418322686181363906, 6.72003899993339773950493315246, 7.05966473800081490091249911134, 7.51701317787040087978951613833, 7.59608287421414082847734504070, 8.279918374656114673852381301808, 8.317217060703710671569510490230, 9.141427446431982795287675525591, 9.311510540105427808001387077179, 9.832189284610846138830258949718
