| L(s) = 1 | + 2.41·2-s − 3-s + 3.82·4-s − 2.41·6-s + 7-s + 4.41·8-s + 9-s + 2·11-s − 3.82·12-s − 13-s + 2.41·14-s + 2.99·16-s − 0.828·17-s + 2.41·18-s − 5.65·19-s − 21-s + 4.82·22-s + 1.17·23-s − 4.41·24-s − 5·25-s − 2.41·26-s − 27-s + 3.82·28-s − 3.65·29-s + 1.65·31-s − 1.58·32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.577·3-s + 1.91·4-s − 0.985·6-s + 0.377·7-s + 1.56·8-s + 0.333·9-s + 0.603·11-s − 1.10·12-s − 0.277·13-s + 0.645·14-s + 0.749·16-s − 0.200·17-s + 0.569·18-s − 1.29·19-s − 0.218·21-s + 1.02·22-s + 0.244·23-s − 0.901·24-s − 25-s − 0.473·26-s − 0.192·27-s + 0.723·28-s − 0.679·29-s + 0.297·31-s − 0.280·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.698665159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.698665159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 0.343T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−12.11879618305232732790366468940, −11.29518608843858614815163999582, −10.55765662806467945537028057206, −9.083653609151694394209207435823, −7.54413780415935792002410909489, −6.50523871037060419535490232534, −5.71245602426191992357572839372, −4.63869215328836944881622050919, −3.81564192422094497201229732335, −2.14157049850050474625482236726,
2.14157049850050474625482236726, 3.81564192422094497201229732335, 4.63869215328836944881622050919, 5.71245602426191992357572839372, 6.50523871037060419535490232534, 7.54413780415935792002410909489, 9.083653609151694394209207435823, 10.55765662806467945537028057206, 11.29518608843858614815163999582, 12.11879618305232732790366468940
