| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 3·7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 3·14-s + 16-s − 3·17-s + 18-s + 3·19-s − 3·21-s + 22-s − 23-s − 24-s − 5·25-s + 26-s − 27-s + 3·28-s − 4·29-s − 6·31-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.688·19-s − 0.654·21-s + 0.213·22-s − 0.208·23-s − 0.204·24-s − 25-s + 0.196·26-s − 0.192·27-s + 0.566·28-s − 0.742·29-s − 1.07·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680807827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680807827\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.13468436876908831792790809466, −11.39416312946457166463535539744, −10.78551225655244359412963255999, −9.435475333424262584223799187666, −8.093256336428860279579050698932, −7.07433016066728121125499972487, −5.85671815365648243578194470848, −4.93990077281798148757107408401, −3.81933209527513554138512512650, −1.81998707337995438672490638604,
1.81998707337995438672490638604, 3.81933209527513554138512512650, 4.93990077281798148757107408401, 5.85671815365648243578194470848, 7.07433016066728121125499972487, 8.093256336428860279579050698932, 9.435475333424262584223799187666, 10.78551225655244359412963255999, 11.39416312946457166463535539744, 12.13468436876908831792790809466
