| L(s) = 1 | + 2.70·2-s − 3-s + 5.30·4-s + 2.79·5-s − 2.70·6-s + 0.880·7-s + 8.93·8-s + 9-s + 7.56·10-s − 5.67·11-s − 5.30·12-s − 3.71·13-s + 2.38·14-s − 2.79·15-s + 13.5·16-s + 4.80·17-s + 2.70·18-s + 6.68·19-s + 14.8·20-s − 0.880·21-s − 15.3·22-s + 23-s − 8.93·24-s + 2.83·25-s − 10.0·26-s − 27-s + 4.67·28-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.65·4-s + 1.25·5-s − 1.10·6-s + 0.332·7-s + 3.16·8-s + 0.333·9-s + 2.39·10-s − 1.71·11-s − 1.53·12-s − 1.03·13-s + 0.636·14-s − 0.722·15-s + 3.38·16-s + 1.16·17-s + 0.637·18-s + 1.53·19-s + 3.32·20-s − 0.192·21-s − 3.27·22-s + 0.208·23-s − 1.82·24-s + 0.567·25-s − 1.97·26-s − 0.192·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.119653060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.119653060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 - 0.880T + 7T^{2} \) |
| 11 | \( 1 + 5.67T + 11T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 31 | \( 1 - 0.779T + 31T^{2} \) |
| 37 | \( 1 + 8.14T + 37T^{2} \) |
| 41 | \( 1 - 6.06T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 + 7.68T + 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 + 7.99T + 61T^{2} \) |
| 67 | \( 1 + 8.36T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 0.302T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 0.884T + 97T^{2} \) |
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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
−9.593008816505147064421878980863, −7.73981341510030399102271249558, −7.44646219567361043376682494011, −6.35964053758665111249673470745, −5.49134484962252314845070551961, −5.32603434154664824844106737998, −4.68015582971394297989493895834, −3.24012385766660849629404172801, −2.56633773201439682564786391001, −1.55430180382259652864524918958,
1.55430180382259652864524918958, 2.56633773201439682564786391001, 3.24012385766660849629404172801, 4.68015582971394297989493895834, 5.32603434154664824844106737998, 5.49134484962252314845070551961, 6.35964053758665111249673470745, 7.44646219567361043376682494011, 7.73981341510030399102271249558, 9.593008816505147064421878980863
