L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s + 14-s + 16-s − 18-s + 21-s − 3·22-s − 23-s + 24-s − 27-s − 28-s − 2·29-s − 10·31-s − 32-s − 3·33-s + 36-s − 37-s − 10·41-s − 42-s + 9·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 1.79·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s − 0.164·37-s − 1.56·41-s − 0.154·42-s + 1.37·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.91810489252341, −12.56222294803550, −12.17861855731462, −11.64504223529918, −11.32585328207654, −10.78166186690834, −10.42223692335304, −9.967957818417276, −9.276502533950216, −9.183008937345441, −8.694503958295357, −7.993509269808423, −7.497161700619942, −7.095694681118775, −6.694340717210193, −5.997702529582832, −5.868465254678430, −5.210569821029681, −4.477781072231137, −4.101579514936166, −3.348432994349360, −3.046653222991606, −2.108783571730069, −1.571411130691194, −1.173638942692034, 0, 0,
1.173638942692034, 1.571411130691194, 2.108783571730069, 3.046653222991606, 3.348432994349360, 4.101579514936166, 4.477781072231137, 5.210569821029681, 5.868465254678430, 5.997702529582832, 6.694340717210193, 7.095694681118775, 7.497161700619942, 7.993509269808423, 8.694503958295357, 9.183008937345441, 9.276502533950216, 9.967957818417276, 10.42223692335304, 10.78166186690834, 11.32585328207654, 11.64504223529918, 12.17861855731462, 12.56222294803550, 12.91810489252341