| L(s) = 1 | + 3-s − 9-s + 2·11-s − 5·13-s − 2·17-s + 2·19-s + 2·23-s + 3·29-s − 9·31-s + 2·33-s − 5·39-s + 41-s + 16·43-s − 11·47-s − 14·49-s − 2·51-s − 4·53-s + 2·57-s + 4·59-s + 8·61-s + 2·67-s + 2·69-s + 23·71-s + 17·73-s − 2·79-s − 4·81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/3·9-s + 0.603·11-s − 1.38·13-s − 0.485·17-s + 0.458·19-s + 0.417·23-s + 0.557·29-s − 1.61·31-s + 0.348·33-s − 0.800·39-s + 0.156·41-s + 2.43·43-s − 1.60·47-s − 2·49-s − 0.280·51-s − 0.549·53-s + 0.264·57-s + 0.520·59-s + 1.02·61-s + 0.244·67-s + 0.240·69-s + 2.72·71-s + 1.98·73-s − 0.225·79-s − 4/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.371433361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.371433361\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 23 T + 270 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.459315617046651073932519656180, −8.282260236464406159746418682264, −7.60098075350262837683105282923, −7.50940527140274791465455324565, −7.21756114267031494867997027210, −6.66152523171619072891076458678, −6.25251504389094213992456101703, −6.24398949123731821073498930327, −5.29980570948004101491612230503, −5.17774338010131377346750686654, −4.95684830201100696518791760752, −4.34679277085415124609656291700, −3.87279828490676008199182221474, −3.61946316161355242642462157927, −2.95175399201063850692075784139, −2.80778781793794385017375465096, −2.08232035691823777069841527985, −1.94983174031447491261170566818, −1.08003449489456573902734313339, −0.43718609551609084021564637761,
0.43718609551609084021564637761, 1.08003449489456573902734313339, 1.94983174031447491261170566818, 2.08232035691823777069841527985, 2.80778781793794385017375465096, 2.95175399201063850692075784139, 3.61946316161355242642462157927, 3.87279828490676008199182221474, 4.34679277085415124609656291700, 4.95684830201100696518791760752, 5.17774338010131377346750686654, 5.29980570948004101491612230503, 6.24398949123731821073498930327, 6.25251504389094213992456101703, 6.66152523171619072891076458678, 7.21756114267031494867997027210, 7.50940527140274791465455324565, 7.60098075350262837683105282923, 8.282260236464406159746418682264, 8.459315617046651073932519656180
