L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 3·8-s + 9-s + 12-s − 4·13-s + 2·14-s − 16-s + 6·17-s + 18-s − 6·19-s − 2·21-s − 4·23-s + 3·24-s − 4·26-s − 27-s − 2·28-s + 6·29-s + 8·31-s + 5·32-s + 6·34-s − 36-s + 6·37-s − 6·38-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 0.534·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.37·19-s − 0.436·21-s − 0.834·23-s + 0.612·24-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + 1.43·31-s + 0.883·32-s + 1.02·34-s − 1/6·36-s + 0.986·37-s − 0.973·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.41475321437809035491184492202, −6.28430969819310006189510844617, −6.08598386330031551698826788433, −4.99783150819232222291640223925, −4.79078422918902924777600712566, −4.13411067705958410301366694387, −3.17569966876548118518813908856, −2.32161313475227573311788760930, −1.15041488736547096377700755888, 0,
1.15041488736547096377700755888, 2.32161313475227573311788760930, 3.17569966876548118518813908856, 4.13411067705958410301366694387, 4.79078422918902924777600712566, 4.99783150819232222291640223925, 6.08598386330031551698826788433, 6.28430969819310006189510844617, 7.41475321437809035491184492202