| L(s) = 1 | + 2-s + 3-s + 4-s − 3.20·5-s + 6-s + 8-s + 9-s − 3.20·10-s + 11-s + 12-s − 6.40·13-s − 3.20·15-s + 16-s + 17-s + 18-s + 1.15·19-s − 3.20·20-s + 22-s + 1.95·23-s + 24-s + 5.24·25-s − 6.40·26-s + 27-s + 7.24·29-s − 3.20·30-s + 10.4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.43·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.01·10-s + 0.301·11-s + 0.288·12-s − 1.77·13-s − 0.826·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.264·19-s − 0.715·20-s + 0.213·22-s + 0.407·23-s + 0.204·24-s + 1.04·25-s − 1.25·26-s + 0.192·27-s + 1.34·29-s − 0.584·30-s + 1.88·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.693645736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.693645736\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.20T + 5T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 1.15T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 5.15T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 5.15T + 43T^{2} \) |
| 47 | \( 1 + 6.04T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 + 2.09T + 73T^{2} \) |
| 79 | \( 1 + 5.60T + 79T^{2} \) |
| 83 | \( 1 - 6.55T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
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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
−8.211754249585844982403404457580, −8.024618379774334432069672158956, −7.11610568533947011796601286172, −6.66708073945498666020429551453, −5.37439856560834501477387514348, −4.49694838415262995135125107964, −4.15127185409954840313256277676, −3.01704775624690576312803390375, −2.54023504065280573334485791586, −0.862789511452685067296126201505,
0.862789511452685067296126201505, 2.54023504065280573334485791586, 3.01704775624690576312803390375, 4.15127185409954840313256277676, 4.49694838415262995135125107964, 5.37439856560834501477387514348, 6.66708073945498666020429551453, 7.11610568533947011796601286172, 8.024618379774334432069672158956, 8.211754249585844982403404457580
