| L(s) = 1 | + 2.23·2-s + 3-s + 3.00·4-s + 1.57·5-s + 2.23·6-s − 1.14·7-s + 2.24·8-s + 9-s + 3.51·10-s − 1.30·11-s + 3.00·12-s + 3.08·13-s − 2.55·14-s + 1.57·15-s − 0.988·16-s + 0.952·17-s + 2.23·18-s + 8.67·19-s + 4.72·20-s − 1.14·21-s − 2.91·22-s + 23-s + 2.24·24-s − 2.52·25-s + 6.89·26-s + 27-s − 3.42·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.703·5-s + 0.913·6-s − 0.431·7-s + 0.793·8-s + 0.333·9-s + 1.11·10-s − 0.392·11-s + 0.866·12-s + 0.854·13-s − 0.681·14-s + 0.406·15-s − 0.247·16-s + 0.231·17-s + 0.527·18-s + 1.98·19-s + 1.05·20-s − 0.248·21-s − 0.620·22-s + 0.208·23-s + 0.457·24-s − 0.504·25-s + 1.35·26-s + 0.192·27-s − 0.647·28-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
| \(L(1)\) |
\(\approx\) |
\(5.856879223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.856879223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 1.57T + 5T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 0.952T + 17T^{2} \) |
| 19 | \( 1 - 8.67T + 19T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 + 0.355T + 37T^{2} \) |
| 41 | \( 1 + 0.628T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 7.28T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.360537437740227978391906180730, −8.259644440777914641769842124071, −7.36914311950825775787535216240, −6.51655958008563974004552952978, −5.77822857526982375195917115561, −5.18926354155385086279929674877, −4.17676545168380986687401722664, −3.25857984271738725333351661470, −2.76309687225205355522115749324, −1.48675419053447724075631121365,
1.48675419053447724075631121365, 2.76309687225205355522115749324, 3.25857984271738725333351661470, 4.17676545168380986687401722664, 5.18926354155385086279929674877, 5.77822857526982375195917115561, 6.51655958008563974004552952978, 7.36914311950825775787535216240, 8.259644440777914641769842124071, 9.360537437740227978391906180730
