L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s − 2·7-s + 8-s + 9-s − 3·10-s − 6·11-s + 12-s − 2·14-s − 3·15-s + 16-s − 3·17-s + 18-s − 2·19-s − 3·20-s − 2·21-s − 6·22-s − 6·23-s + 24-s + 4·25-s + 27-s − 2·28-s + 3·29-s − 3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 1.80·11-s + 0.288·12-s − 0.534·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.458·19-s − 0.670·20-s − 0.436·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.192·27-s − 0.377·28-s + 0.557·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.685758291488366149210341725320, −8.191317221559137847971361455915, −8.095475412748459834666754746001, −7.04629587770525720046073855524, −6.16684971754953097309217392338, −4.90273853094022032181335062300, −4.13479061438517346013965712432, −3.20396144410593418876019292818, −2.39404286061527545470700387762, 0,
2.39404286061527545470700387762, 3.20396144410593418876019292818, 4.13479061438517346013965712432, 4.90273853094022032181335062300, 6.16684971754953097309217392338, 7.04629587770525720046073855524, 8.095475412748459834666754746001, 8.191317221559137847971361455915, 9.685758291488366149210341725320