| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 6·5-s − 4·6-s + 4·8-s + 3·9-s − 12·10-s − 2·11-s − 6·12-s + 12·15-s + 5·16-s + 6·18-s + 2·19-s − 18·20-s − 4·22-s + 4·23-s − 8·24-s + 19·25-s − 4·27-s + 10·29-s + 24·30-s − 10·31-s + 6·32-s + 4·33-s + 9·36-s − 8·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 2.68·5-s − 1.63·6-s + 1.41·8-s + 9-s − 3.79·10-s − 0.603·11-s − 1.73·12-s + 3.09·15-s + 5/4·16-s + 1.41·18-s + 0.458·19-s − 4.02·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s + 19/5·25-s − 0.769·27-s + 1.85·29-s + 4.38·30-s − 1.79·31-s + 1.06·32-s + 0.696·33-s + 3/2·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 99 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 282 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 159 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 241 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.69455050728402810596011300212, −7.38556353336312878038613447465, −7.20376195431669540433880207836, −7.07234083213947470372541896923, −6.47227086400524367552556999968, −6.15748150583024484928937796812, −5.67123041018279430921433965550, −5.29816877415669655321696242430, −4.89562429515684703647924779871, −4.80805777510636754975723185551, −4.14077065403950660365487494320, −4.13543208222462487273065440528, −3.50183375947912428865665553050, −3.42342812643037481036938482380, −2.73363027048482320977641619538, −2.48467441877462232304901553534, −1.35857257526470938939943373715, −1.14098873665669935040871863935, 0, 0,
1.14098873665669935040871863935, 1.35857257526470938939943373715, 2.48467441877462232304901553534, 2.73363027048482320977641619538, 3.42342812643037481036938482380, 3.50183375947912428865665553050, 4.13543208222462487273065440528, 4.14077065403950660365487494320, 4.80805777510636754975723185551, 4.89562429515684703647924779871, 5.29816877415669655321696242430, 5.67123041018279430921433965550, 6.15748150583024484928937796812, 6.47227086400524367552556999968, 7.07234083213947470372541896923, 7.20376195431669540433880207836, 7.38556353336312878038613447465, 7.69455050728402810596011300212
