L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 3·11-s − 12-s + 13-s − 14-s + 16-s − 2.53·17-s − 18-s − 21-s − 3·22-s + 1.53·23-s + 24-s − 5·25-s − 26-s − 27-s + 28-s + 4·31-s − 32-s − 3·33-s + 2.53·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 0.250·16-s − 0.613·17-s − 0.235·18-s − 0.218·21-s − 0.639·22-s + 0.319·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.718·31-s − 0.176·32-s − 0.522·33-s + 0.434·34-s + 0.166·36-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 - 3.53T + 41T^{2} \) |
| 43 | \( 1 - 3.53T + 43T^{2} \) |
| 47 | \( 1 + 0.468T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 1.06T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 + 9.06T + 79T^{2} \) |
| 83 | \( 1 - 9.06T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.51904360147539411559254789457, −6.76436811596664401302268813127, −6.26437069808762751728525774063, −5.56340933589476767118856649691, −4.64714249185143162519713610086, −3.99088063152956238308411010891, −3.01128590005100098459338882481, −1.89061483785317462751937237823, −1.21640907719390999594714792637, 0,
1.21640907719390999594714792637, 1.89061483785317462751937237823, 3.01128590005100098459338882481, 3.99088063152956238308411010891, 4.64714249185143162519713610086, 5.56340933589476767118856649691, 6.26437069808762751728525774063, 6.76436811596664401302268813127, 7.51904360147539411559254789457