| L(s) = 1 | − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s + 4·11-s + 6·13-s − 4·14-s − 16-s − 6·17-s − 4·19-s + 20-s − 4·22-s + 4·23-s + 25-s − 6·26-s − 4·28-s − 29-s − 8·31-s − 5·32-s + 6·34-s − 4·35-s + 2·37-s + 4·38-s − 3·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 0.755·28-s − 0.185·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 0.676·35-s + 0.328·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203478369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203478369\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.230470298962862784846961355987, −8.788063578371319286656665164127, −8.346606260262100304740118719484, −7.41886368904450739537189724511, −6.53264436370468127763928142197, −5.35775714661900019913688256525, −4.23882127299780961454300589319, −3.97331207281193943578109005786, −1.93438057192777317348195684356, −0.984285408377975149151993884914,
0.984285408377975149151993884914, 1.93438057192777317348195684356, 3.97331207281193943578109005786, 4.23882127299780961454300589319, 5.35775714661900019913688256525, 6.53264436370468127763928142197, 7.41886368904450739537189724511, 8.346606260262100304740118719484, 8.788063578371319286656665164127, 9.230470298962862784846961355987
