L(s) = 1 | − 2-s + 4-s + 0.193·7-s − 8-s + 0.514·11-s + 3.12·13-s − 0.193·14-s + 16-s − 2.83·17-s + 19-s − 0.514·22-s + 2.32·23-s − 3.12·26-s + 0.193·28-s − 0.164·29-s − 9.05·31-s − 32-s + 2.83·34-s + 3.02·37-s − 38-s − 9.96·41-s + 5.51·43-s + 0.514·44-s − 2.32·46-s + 1.70·47-s − 6.96·49-s + 3.12·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.0730·7-s − 0.353·8-s + 0.155·11-s + 0.867·13-s − 0.0516·14-s + 0.250·16-s − 0.687·17-s + 0.229·19-s − 0.109·22-s + 0.483·23-s − 0.613·26-s + 0.0365·28-s − 0.0306·29-s − 1.62·31-s − 0.176·32-s + 0.486·34-s + 0.497·37-s − 0.162·38-s − 1.55·41-s + 0.840·43-s + 0.0775·44-s − 0.342·46-s + 0.249·47-s − 0.994·49-s + 0.433·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 0.193T + 7T^{2} \) |
| 11 | \( 1 - 0.514T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 23 | \( 1 - 2.32T + 23T^{2} \) |
| 29 | \( 1 + 0.164T + 29T^{2} \) |
| 31 | \( 1 + 9.05T + 31T^{2} \) |
| 37 | \( 1 - 3.02T + 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 2.90T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 5.92T + 61T^{2} \) |
| 67 | \( 1 - 4.22T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 8.37T + 73T^{2} \) |
| 79 | \( 1 - 6.02T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 9.01T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
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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
−7.49734894417126280544999708495, −6.83499615096798029044322406263, −6.20692760963056923534716117060, −5.49286259782579074807624488413, −4.63883174134705304002475077226, −3.73706315503246543026594160365, −3.03418968439745393946251406913, −1.99014085246750888390597324318, −1.24251477423463550303200139421, 0,
1.24251477423463550303200139421, 1.99014085246750888390597324318, 3.03418968439745393946251406913, 3.73706315503246543026594160365, 4.63883174134705304002475077226, 5.49286259782579074807624488413, 6.20692760963056923534716117060, 6.83499615096798029044322406263, 7.49734894417126280544999708495