L(s) = 1 | + 2-s + 2.04·3-s + 4-s + 3.54·5-s + 2.04·6-s − 0.465·7-s + 8-s + 1.19·9-s + 3.54·10-s − 3.73·11-s + 2.04·12-s + 0.365·13-s − 0.465·14-s + 7.26·15-s + 16-s + 5.28·17-s + 1.19·18-s + 19-s + 3.54·20-s − 0.954·21-s − 3.73·22-s + 1.96·23-s + 2.04·24-s + 7.56·25-s + 0.365·26-s − 3.69·27-s − 0.465·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.58·5-s + 0.836·6-s − 0.176·7-s + 0.353·8-s + 0.398·9-s + 1.12·10-s − 1.12·11-s + 0.591·12-s + 0.101·13-s − 0.124·14-s + 1.87·15-s + 0.250·16-s + 1.28·17-s + 0.281·18-s + 0.229·19-s + 0.792·20-s − 0.208·21-s − 0.796·22-s + 0.409·23-s + 0.418·24-s + 1.51·25-s + 0.0717·26-s − 0.711·27-s − 0.0880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.974217164\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.974217164\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 + 0.465T + 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 - 0.365T + 13T^{2} \) |
| 17 | \( 1 - 5.28T + 17T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 - 6.23T + 31T^{2} \) |
| 37 | \( 1 - 6.00T + 37T^{2} \) |
| 41 | \( 1 + 0.0824T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 0.0108T + 53T^{2} \) |
| 59 | \( 1 - 9.66T + 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 1.59T + 97T^{2} \) |
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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
−7.947422226086644429939481060813, −7.09146653886813632696787548003, −6.33838550882393833567686964259, −5.57189493440533541804250731574, −5.25100325839880116226228084434, −4.24728244397254808764661061989, −3.11632024393966279438951769918, −2.84040429874005420043517490865, −2.13132896880792720991560110010, −1.19180184761112175763942282078,
1.19180184761112175763942282078, 2.13132896880792720991560110010, 2.84040429874005420043517490865, 3.11632024393966279438951769918, 4.24728244397254808764661061989, 5.25100325839880116226228084434, 5.57189493440533541804250731574, 6.33838550882393833567686964259, 7.09146653886813632696787548003, 7.947422226086644429939481060813