| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s − 3·8-s + 9-s + 2·10-s − 12-s + 6·13-s + 2·15-s − 16-s + 2·17-s + 18-s + 4·19-s − 2·20-s − 3·24-s − 25-s + 6·26-s + 27-s + 2·29-s + 2·30-s − 8·31-s + 5·32-s + 2·34-s − 36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s + 1.66·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s + 0.883·32-s + 0.342·34-s − 1/6·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.777774155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.777774155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.62319049413803, −15.21241018183692, −14.42210723543853, −14.05853389192787, −13.73094271842140, −13.20634631261681, −12.75006045871082, −12.19577870200304, −11.37815305429305, −10.84648784608924, −10.06095794901290, −9.568737666473635, −8.930368746492868, −8.746037781155705, −7.762078793064544, −7.302691118297447, −6.137307596189636, −5.911131043902567, −5.428312648397238, −4.425943581208136, −3.957024262768600, −3.238194326833478, −2.659156875385975, −1.619470596263804, −0.8589803999789431,
0.8589803999789431, 1.619470596263804, 2.659156875385975, 3.238194326833478, 3.957024262768600, 4.425943581208136, 5.428312648397238, 5.911131043902567, 6.137307596189636, 7.302691118297447, 7.762078793064544, 8.746037781155705, 8.930368746492868, 9.568737666473635, 10.06095794901290, 10.84648784608924, 11.37815305429305, 12.19577870200304, 12.75006045871082, 13.20634631261681, 13.73094271842140, 14.05853389192787, 14.42210723543853, 15.21241018183692, 15.62319049413803
