| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 5.41·11-s + 12-s + 2.82·13-s + 16-s − 6.82·17-s + 18-s − 19-s − 5.41·22-s + 4.24·23-s + 24-s − 5·25-s + 2.82·26-s + 27-s − 8·29-s − 7.65·31-s + 32-s − 5.41·33-s − 6.82·34-s + 36-s − 2·37-s − 38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.63·11-s + 0.288·12-s + 0.784·13-s + 0.250·16-s − 1.65·17-s + 0.235·18-s − 0.229·19-s − 1.15·22-s + 0.884·23-s + 0.204·24-s − 25-s + 0.554·26-s + 0.192·27-s − 1.48·29-s − 1.37·31-s + 0.176·32-s − 0.942·33-s − 1.17·34-s + 0.166·36-s − 0.328·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 + 1.17T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.41T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 1.65T + 83T^{2} \) |
| 89 | \( 1 - 5.41T + 89T^{2} \) |
| 97 | \( 1 - 9.17T + 97T^{2} \) |
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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
−7.60283762747182366824230729549, −7.20333305202333777091746628807, −6.27812838527442566130070108129, −5.51720329725802495842378503119, −4.87967367283334255145564842920, −3.99206389457663419228151005707, −3.34315470028651652947309795224, −2.39764942310117396721667647527, −1.80168911370043170760863322425, 0,
1.80168911370043170760863322425, 2.39764942310117396721667647527, 3.34315470028651652947309795224, 3.99206389457663419228151005707, 4.87967367283334255145564842920, 5.51720329725802495842378503119, 6.27812838527442566130070108129, 7.20333305202333777091746628807, 7.60283762747182366824230729549
