L(s) = 1 | − 2·3-s + 3·9-s + 4·11-s + 4·13-s − 8·17-s − 4·23-s + 2·25-s − 4·27-s − 4·29-s − 8·31-s − 8·33-s + 4·37-s − 8·39-s − 16·41-s − 16·43-s + 8·47-s + 16·51-s + 4·53-s + 16·59-s − 4·61-s + 8·67-s + 8·69-s − 12·71-s − 12·73-s − 4·75-s + 8·79-s + 5·81-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s + 1.20·11-s + 1.10·13-s − 1.94·17-s − 0.834·23-s + 2/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s − 1.39·33-s + 0.657·37-s − 1.28·39-s − 2.49·41-s − 2.43·43-s + 1.16·47-s + 2.24·51-s + 0.549·53-s + 2.08·59-s − 0.512·61-s + 0.977·67-s + 0.963·69-s − 1.42·71-s − 1.40·73-s − 0.461·75-s + 0.900·79-s + 5/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.11457556150614888002847459674, −7.09536912984812111731608418318, −6.87223295395087364552356383330, −6.54452745151342578597740221740, −6.11496701437998752436670479739, −5.98047010335366169344027546632, −5.42394772179762228906445326732, −5.23708612647897533857359546153, −4.79273265910832254025447315884, −4.39302846430666336361843953328, −3.92007417955532563623689185237, −3.84969312264330882203501203089, −3.45821801223985192203586797087, −2.87282502434015661290993280669, −2.02647888119039871467488963636, −2.01741251832874320211315377965, −1.40951223292203075536820948492, −1.01750916340233385417297320711, 0, 0,
1.01750916340233385417297320711, 1.40951223292203075536820948492, 2.01741251832874320211315377965, 2.02647888119039871467488963636, 2.87282502434015661290993280669, 3.45821801223985192203586797087, 3.84969312264330882203501203089, 3.92007417955532563623689185237, 4.39302846430666336361843953328, 4.79273265910832254025447315884, 5.23708612647897533857359546153, 5.42394772179762228906445326732, 5.98047010335366169344027546632, 6.11496701437998752436670479739, 6.54452745151342578597740221740, 6.87223295395087364552356383330, 7.09536912984812111731608418318, 7.11457556150614888002847459674