| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s − 14-s − 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s + 21-s − 3·22-s − 23-s − 24-s − 4·25-s − 27-s − 28-s + 29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.639·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.188·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.110246398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.110246398\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.21758809262731, −12.83908534040016, −12.34344821541264, −11.86704698293373, −11.54132535637180, −10.76601689086256, −10.64851807540252, −9.852182279594295, −9.645834788682100, −9.189849843898492, −8.144507807362284, −7.797590074485492, −7.551768283015912, −6.525673472046451, −6.434212768349277, −5.785286215766103, −5.441711662266848, −4.816601995019435, −4.435109771475533, −3.734005022011203, −2.970378391214611, −2.756493378282047, −1.907114446671336, −1.238076276197848, −0.4831718910383431,
0.4831718910383431, 1.238076276197848, 1.907114446671336, 2.756493378282047, 2.970378391214611, 3.734005022011203, 4.435109771475533, 4.816601995019435, 5.441711662266848, 5.785286215766103, 6.434212768349277, 6.525673472046451, 7.551768283015912, 7.797590074485492, 8.144507807362284, 9.189849843898492, 9.645834788682100, 9.852182279594295, 10.64851807540252, 10.76601689086256, 11.54132535637180, 11.86704698293373, 12.34344821541264, 12.83908534040016, 13.21758809262731
