L(s) = 1 | + 2-s + 0.706·3-s + 4-s + 0.706·6-s − 2.40·7-s + 8-s − 2.50·9-s + 4.95·11-s + 0.706·12-s + 4.40·13-s − 2.40·14-s + 16-s − 0.200·17-s − 2.50·18-s + 6.65·19-s − 1.70·21-s + 4.95·22-s + 23-s + 0.706·24-s + 4.40·26-s − 3.88·27-s − 2.40·28-s + 1.67·29-s − 1.82·31-s + 32-s + 3.50·33-s − 0.200·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.407·3-s + 0.5·4-s + 0.288·6-s − 0.910·7-s + 0.353·8-s − 0.833·9-s + 1.49·11-s + 0.203·12-s + 1.22·13-s − 0.643·14-s + 0.250·16-s − 0.0487·17-s − 0.589·18-s + 1.52·19-s − 0.371·21-s + 1.05·22-s + 0.208·23-s + 0.144·24-s + 0.864·26-s − 0.748·27-s − 0.455·28-s + 0.311·29-s − 0.327·31-s + 0.176·32-s + 0.609·33-s − 0.0344·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.891408604\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.891408604\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 0.706T + 3T^{2} \) |
| 7 | \( 1 + 2.40T + 7T^{2} \) |
| 11 | \( 1 - 4.95T + 11T^{2} \) |
| 13 | \( 1 - 4.40T + 13T^{2} \) |
| 17 | \( 1 + 0.200T + 17T^{2} \) |
| 19 | \( 1 - 6.65T + 19T^{2} \) |
| 29 | \( 1 - 1.67T + 29T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 + 0.505T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 + 6.67T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 5.04T + 89T^{2} \) |
| 97 | \( 1 + 3.79T + 97T^{2} \) |
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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
−9.481745513631826084891734137914, −9.212002281946319828218329999616, −8.146383851894313246654591852957, −7.16807273780554223021201214583, −6.18136965873921016182702735043, −5.83733296624171783265238845461, −4.39901784012987546315663699004, −3.47626542437341727155408012057, −2.91055280880067350922482125998, −1.27211587416571166300629431866,
1.27211587416571166300629431866, 2.91055280880067350922482125998, 3.47626542437341727155408012057, 4.39901784012987546315663699004, 5.83733296624171783265238845461, 6.18136965873921016182702735043, 7.16807273780554223021201214583, 8.146383851894313246654591852957, 9.212002281946319828218329999616, 9.481745513631826084891734137914