L(s) = 1 | + 2-s + 2.18·3-s + 4-s + 5-s + 2.18·6-s − 7-s + 8-s + 1.76·9-s + 10-s + 2.18·12-s − 2.53·13-s − 14-s + 2.18·15-s + 16-s + 5.73·17-s + 1.76·18-s + 2.24·19-s + 20-s − 2.18·21-s − 2.51·23-s + 2.18·24-s + 25-s − 2.53·26-s − 2.69·27-s − 28-s + 8.61·29-s + 2.18·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s + 0.447·5-s + 0.891·6-s − 0.377·7-s + 0.353·8-s + 0.588·9-s + 0.316·10-s + 0.630·12-s − 0.702·13-s − 0.267·14-s + 0.563·15-s + 0.250·16-s + 1.39·17-s + 0.416·18-s + 0.514·19-s + 0.223·20-s − 0.476·21-s − 0.524·23-s + 0.445·24-s + 0.200·25-s − 0.496·26-s − 0.518·27-s − 0.188·28-s + 1.60·29-s + 0.398·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.878583527\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.878583527\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 2.51T + 23T^{2} \) |
| 29 | \( 1 - 8.61T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 2.18T + 43T^{2} \) |
| 47 | \( 1 - 3.22T + 47T^{2} \) |
| 53 | \( 1 + 4.01T + 53T^{2} \) |
| 59 | \( 1 - 2.01T + 59T^{2} \) |
| 61 | \( 1 - 0.268T + 61T^{2} \) |
| 67 | \( 1 - 3.86T + 67T^{2} \) |
| 71 | \( 1 + 9.55T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 7.34T + 83T^{2} \) |
| 89 | \( 1 + 2.65T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.80370986418016181202997338119, −7.18163080305679110838276200282, −6.36282338987220644609203917116, −5.66687815858713794206763972541, −4.98012570500814712533038360704, −4.09207224205577870995241042793, −3.34337134544425258835176913886, −2.77541316228050721768010033430, −2.16967715942033118624295863563, −1.03338901485360479204673780015,
1.03338901485360479204673780015, 2.16967715942033118624295863563, 2.77541316228050721768010033430, 3.34337134544425258835176913886, 4.09207224205577870995241042793, 4.98012570500814712533038360704, 5.66687815858713794206763972541, 6.36282338987220644609203917116, 7.18163080305679110838276200282, 7.80370986418016181202997338119