| L(s) = 1 | − 0.792·2-s + 2.09·3-s − 1.37·4-s − 1.66·6-s + 7-s + 2.67·8-s + 1.40·9-s − 3.35·11-s − 2.88·12-s + 0.369·13-s − 0.792·14-s + 0.626·16-s − 4.14·17-s − 1.11·18-s + 7.06·19-s + 2.09·21-s + 2.66·22-s − 23-s + 5.61·24-s − 0.292·26-s − 3.34·27-s − 1.37·28-s − 1.49·29-s − 9.45·31-s − 5.84·32-s − 7.04·33-s + 3.28·34-s + ⋯ |
| L(s) = 1 | − 0.560·2-s + 1.21·3-s − 0.686·4-s − 0.679·6-s + 0.377·7-s + 0.944·8-s + 0.469·9-s − 1.01·11-s − 0.831·12-s + 0.102·13-s − 0.211·14-s + 0.156·16-s − 1.00·17-s − 0.262·18-s + 1.62·19-s + 0.458·21-s + 0.567·22-s − 0.208·23-s + 1.14·24-s − 0.0574·26-s − 0.643·27-s − 0.259·28-s − 0.276·29-s − 1.69·31-s − 1.03·32-s − 1.22·33-s + 0.563·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.792T + 2T^{2} \) |
| 3 | \( 1 - 2.09T + 3T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 0.369T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 + 9.45T + 31T^{2} \) |
| 37 | \( 1 + 4.65T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 + 0.208T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 - 2.53T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 0.186T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.114695418925455058989459052988, −7.67204150919360586040108329148, −7.08150074140287152869245668012, −5.62872896696679734131952751420, −5.08328783514639112568204751495, −4.10377766651320032698901362958, −3.36226073878227216832240443419, −2.42109358369595514934554568569, −1.49537533645777060959635426657, 0,
1.49537533645777060959635426657, 2.42109358369595514934554568569, 3.36226073878227216832240443419, 4.10377766651320032698901362958, 5.08328783514639112568204751495, 5.62872896696679734131952751420, 7.08150074140287152869245668012, 7.67204150919360586040108329148, 8.114695418925455058989459052988
