| L(s) = 1 | − 2-s − 3-s + 4-s + 3.59·5-s + 6-s + 0.919·7-s − 8-s + 9-s − 3.59·10-s + 6.28·11-s − 12-s − 13-s − 0.919·14-s − 3.59·15-s + 16-s − 0.964·17-s − 18-s + 5.87·19-s + 3.59·20-s − 0.919·21-s − 6.28·22-s − 1.55·23-s + 24-s + 7.95·25-s + 26-s − 27-s + 0.919·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.60·5-s + 0.408·6-s + 0.347·7-s − 0.353·8-s + 0.333·9-s − 1.13·10-s + 1.89·11-s − 0.288·12-s − 0.277·13-s − 0.245·14-s − 0.929·15-s + 0.250·16-s − 0.233·17-s − 0.235·18-s + 1.34·19-s + 0.804·20-s − 0.200·21-s − 1.33·22-s − 0.324·23-s + 0.204·24-s + 1.59·25-s + 0.196·26-s − 0.192·27-s + 0.173·28-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
| \(L(1)\) |
\(\approx\) |
\(2.265321457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.265321457\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 0.919T + 7T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 17 | \( 1 + 0.964T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 + 1.55T + 23T^{2} \) |
| 29 | \( 1 - 0.377T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 5.17T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 - 6.01T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 0.838T + 59T^{2} \) |
| 61 | \( 1 - 0.508T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−7.75371248073578767867790617966, −7.04327021663908952244642678262, −6.31949944952989561666434688123, −6.04792589697604768894615913549, −5.20544360258488790739747506196, −4.45689742109079549376901946329, −3.38287072543190400095278455438, −2.32541213856617441924763562378, −1.51937347645770170240588977204, −0.966650487979642404321199326636,
0.966650487979642404321199326636, 1.51937347645770170240588977204, 2.32541213856617441924763562378, 3.38287072543190400095278455438, 4.45689742109079549376901946329, 5.20544360258488790739747506196, 6.04792589697604768894615913549, 6.31949944952989561666434688123, 7.04327021663908952244642678262, 7.75371248073578767867790617966
