| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s + 2·11-s − 8·14-s + 5·16-s + 4·17-s − 6·19-s − 6·20-s + 4·22-s − 12·28-s − 8·29-s + 8·31-s + 6·32-s + 8·34-s + 8·35-s + 2·37-s − 12·38-s − 8·40-s − 12·41-s − 6·43-s + 6·44-s − 12·47-s − 2·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s − 2.13·14-s + 5/4·16-s + 0.970·17-s − 1.37·19-s − 1.34·20-s + 0.852·22-s − 2.26·28-s − 1.48·29-s + 1.43·31-s + 1.06·32-s + 1.37·34-s + 1.35·35-s + 0.328·37-s − 1.94·38-s − 1.26·40-s − 1.87·41-s − 0.914·43-s + 0.904·44-s − 1.75·47-s − 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90668484 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 16 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 100 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 120 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 122 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 280 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 182 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.24515776565207922372967763351, −7.00250064876291098976375330068, −6.65488293176757595381630936310, −6.62736674283844275989223017120, −6.15259847368495097183841229619, −5.79960713410923103516763302826, −5.54830292795432285644473930720, −5.00910843941488530672493320934, −4.57037203427772143971788940150, −4.51844857312380677350214928843, −3.79788563375317648573894904528, −3.67543712045665851254515389605, −3.26808122581276878806932088356, −3.25689697812431168168648625385, −2.52986271483881353465043552105, −2.17444067041215523977959673401, −1.56327446107957143644344906927, −1.13354214469691637143842240064, 0, 0,
1.13354214469691637143842240064, 1.56327446107957143644344906927, 2.17444067041215523977959673401, 2.52986271483881353465043552105, 3.25689697812431168168648625385, 3.26808122581276878806932088356, 3.67543712045665851254515389605, 3.79788563375317648573894904528, 4.51844857312380677350214928843, 4.57037203427772143971788940150, 5.00910843941488530672493320934, 5.54830292795432285644473930720, 5.79960713410923103516763302826, 6.15259847368495097183841229619, 6.62736674283844275989223017120, 6.65488293176757595381630936310, 7.00250064876291098976375330068, 7.24515776565207922372967763351
