| L(s) = 1 | − 0.970·2-s + 1.98·3-s − 1.05·4-s − 1.92·6-s + 0.240·7-s + 2.96·8-s + 0.950·9-s − 0.146·11-s − 2.10·12-s − 2.69·13-s − 0.233·14-s − 0.766·16-s − 0.922·18-s − 7.91·19-s + 0.478·21-s + 0.142·22-s + 4.85·23-s + 5.89·24-s + 2.61·26-s − 4.07·27-s − 0.254·28-s + 3.57·29-s + 1.39·31-s − 5.19·32-s − 0.291·33-s − 1.00·36-s + 6.86·37-s + ⋯ |
| L(s) = 1 | − 0.686·2-s + 1.14·3-s − 0.528·4-s − 0.787·6-s + 0.0909·7-s + 1.04·8-s + 0.316·9-s − 0.0442·11-s − 0.606·12-s − 0.748·13-s − 0.0624·14-s − 0.191·16-s − 0.217·18-s − 1.81·19-s + 0.104·21-s + 0.0303·22-s + 1.01·23-s + 1.20·24-s + 0.513·26-s − 0.784·27-s − 0.0481·28-s + 0.664·29-s + 0.251·31-s − 0.917·32-s − 0.0508·33-s − 0.167·36-s + 1.12·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.970T + 2T^{2} \) |
| 3 | \( 1 - 1.98T + 3T^{2} \) |
| 7 | \( 1 - 0.240T + 7T^{2} \) |
| 11 | \( 1 + 0.146T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 3.57T + 29T^{2} \) |
| 31 | \( 1 - 1.39T + 31T^{2} \) |
| 37 | \( 1 - 6.86T + 37T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 + 4.44T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 8.35T + 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 - 0.848T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 + 4.82T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−7.77285590680291100469824145887, −7.25398048140197751195625366926, −6.33493342129233232796842811023, −5.38384852219171161275368235808, −4.42134625626803634300976622750, −4.10370349309503564364162359317, −2.87693201030293897307331564921, −2.35803356315171502138437970761, −1.26305507449458588540412241148, 0,
1.26305507449458588540412241148, 2.35803356315171502138437970761, 2.87693201030293897307331564921, 4.10370349309503564364162359317, 4.42134625626803634300976622750, 5.38384852219171161275368235808, 6.33493342129233232796842811023, 7.25398048140197751195625366926, 7.77285590680291100469824145887
