| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 3·13-s + 3·14-s + 15-s + 16-s − 18-s − 2·19-s − 20-s + 3·21-s + 4·22-s + 23-s + 24-s + 25-s − 3·26-s − 27-s − 3·28-s − 7·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.832·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.654·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s − 0.566·28-s − 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1066216901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1066216901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.57801136096861, −11.92278486207393, −11.53361937505061, −10.95773272362841, −10.67739889462299, −10.41819105831476, −9.673004756944223, −9.403231714242792, −8.994331695792708, −8.271600656860684, −7.961722492818813, −7.470341028383260, −7.042545109374107, −6.416981347795413, −6.121316675186645, −5.676611895630074, −5.082302873190352, −4.504267212658414, −3.817458857863619, −3.456581923609485, −2.781931658914958, −2.367795526712798, −1.547663133303884, −0.8947095709783248, −0.1205360841800299,
0.1205360841800299, 0.8947095709783248, 1.547663133303884, 2.367795526712798, 2.781931658914958, 3.456581923609485, 3.817458857863619, 4.504267212658414, 5.082302873190352, 5.676611895630074, 6.121316675186645, 6.416981347795413, 7.042545109374107, 7.470341028383260, 7.961722492818813, 8.271600656860684, 8.994331695792708, 9.403231714242792, 9.673004756944223, 10.41819105831476, 10.67739889462299, 10.95773272362841, 11.53361937505061, 11.92278486207393, 12.57801136096861
