| L(s) = 1 | − 2-s − 3-s + 4-s + 0.522·5-s + 6-s − 2.74·7-s − 8-s + 9-s − 0.522·10-s − 0.360·11-s − 12-s − 13-s + 2.74·14-s − 0.522·15-s + 16-s + 1.93·17-s − 18-s − 0.748·19-s + 0.522·20-s + 2.74·21-s + 0.360·22-s − 1.23·23-s + 24-s − 4.72·25-s + 26-s − 27-s − 2.74·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.233·5-s + 0.408·6-s − 1.03·7-s − 0.353·8-s + 0.333·9-s − 0.165·10-s − 0.108·11-s − 0.288·12-s − 0.277·13-s + 0.734·14-s − 0.134·15-s + 0.250·16-s + 0.468·17-s − 0.235·18-s − 0.171·19-s + 0.116·20-s + 0.599·21-s + 0.0768·22-s − 0.257·23-s + 0.204·24-s − 0.945·25-s + 0.196·26-s − 0.192·27-s − 0.519·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6094699078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6094699078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 0.522T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 + 0.360T + 11T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 + 0.748T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + 2.99T + 37T^{2} \) |
| 41 | \( 1 - 5.55T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 + 0.902T + 47T^{2} \) |
| 53 | \( 1 + 9.38T + 53T^{2} \) |
| 59 | \( 1 - 6.21T + 59T^{2} \) |
| 61 | \( 1 - 6.71T + 61T^{2} \) |
| 67 | \( 1 - 2.28T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 + 2.77T + 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−7.65845301227760232724684860860, −7.26243189796902525724442816632, −6.37617229375705504252745324056, −5.95884899455880869809272441412, −5.28575572002975074706181465021, −4.22602534258585031906609322297, −3.43105457867528214748456355960, −2.54415939567930207918179983038, −1.62110962744451062155198913911, −0.43870123530732401506963949410,
0.43870123530732401506963949410, 1.62110962744451062155198913911, 2.54415939567930207918179983038, 3.43105457867528214748456355960, 4.22602534258585031906609322297, 5.28575572002975074706181465021, 5.95884899455880869809272441412, 6.37617229375705504252745324056, 7.26243189796902525724442816632, 7.65845301227760232724684860860
