L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 3·7-s + 9-s + 2·11-s − 2·12-s + 13-s − 6·14-s − 4·16-s + 2·17-s + 2·18-s − 5·19-s + 3·21-s + 4·22-s + 6·23-s + 2·26-s − 27-s − 6·28-s + 10·29-s − 3·31-s − 8·32-s − 2·33-s + 4·34-s + 2·36-s + 2·37-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 0.277·13-s − 1.60·14-s − 16-s + 0.485·17-s + 0.471·18-s − 1.14·19-s + 0.654·21-s + 0.852·22-s + 1.25·23-s + 0.392·26-s − 0.192·27-s − 1.13·28-s + 1.85·29-s − 0.538·31-s − 1.41·32-s − 0.348·33-s + 0.685·34-s + 1/3·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.402539940\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.402539940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.43096345053950877210365893366, −13.27531306639888978122196489064, −12.60092562784261018149461459363, −11.69351541999948731726714975337, −10.39882094959826449711333942129, −8.979440455637625961269202062502, −6.79375443883521821659151920822, −6.05569215012514529764032801313, −4.62309319523854591570832526017, −3.24466149791278329969031792942,
3.24466149791278329969031792942, 4.62309319523854591570832526017, 6.05569215012514529764032801313, 6.79375443883521821659151920822, 8.979440455637625961269202062502, 10.39882094959826449711333942129, 11.69351541999948731726714975337, 12.60092562784261018149461459363, 13.27531306639888978122196489064, 14.43096345053950877210365893366