| L(s) = 1 | − 0.554·2-s + 1.24·3-s − 1.69·4-s − 4.24·5-s − 0.692·6-s + 1.91·7-s + 2.04·8-s − 1.44·9-s + 2.35·10-s − 4.96·11-s − 2.10·12-s − 4.04·13-s − 1.06·14-s − 5.29·15-s + 2.24·16-s + 0.643·17-s + 0.801·18-s + 0.939·19-s + 7.18·20-s + 2.38·21-s + 2.75·22-s − 6.44·23-s + 2.55·24-s + 13.0·25-s + 2.24·26-s − 5.54·27-s − 3.23·28-s + ⋯ |
| L(s) = 1 | − 0.392·2-s + 0.719·3-s − 0.846·4-s − 1.89·5-s − 0.282·6-s + 0.722·7-s + 0.724·8-s − 0.481·9-s + 0.745·10-s − 1.49·11-s − 0.609·12-s − 1.12·13-s − 0.283·14-s − 1.36·15-s + 0.561·16-s + 0.155·17-s + 0.189·18-s + 0.215·19-s + 1.60·20-s + 0.520·21-s + 0.586·22-s − 1.34·23-s + 0.521·24-s + 2.60·25-s + 0.440·26-s − 1.06·27-s − 0.611·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 + T \) |
| good | 2 | \( 1 + 0.554T + 2T^{2} \) |
| 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 - 0.643T + 17T^{2} \) |
| 19 | \( 1 - 0.939T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 8.50T + 29T^{2} \) |
| 31 | \( 1 + 0.533T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 0.0978T + 41T^{2} \) |
| 43 | \( 1 - 0.0881T + 43T^{2} \) |
| 47 | \( 1 - 1.86T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + 1.45T + 71T^{2} \) |
| 73 | \( 1 - 9.26T + 73T^{2} \) |
| 79 | \( 1 + 9.30T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 7.06T + 89T^{2} \) |
| 97 | \( 1 - 9.75T + 97T^{2} \) |
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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
−11.90326775571520338890277595574, −10.87863539052210386400418692084, −9.823397449572255857425061177538, −8.442068018791966662476418245187, −8.051400053323129594973183350924, −7.51722403244174989414071372235, −5.11786284131625503834768814751, −4.25805992123448871147203069422, −2.90375338754860413094276037382, 0,
2.90375338754860413094276037382, 4.25805992123448871147203069422, 5.11786284131625503834768814751, 7.51722403244174989414071372235, 8.051400053323129594973183350924, 8.442068018791966662476418245187, 9.823397449572255857425061177538, 10.87863539052210386400418692084, 11.90326775571520338890277595574
