| L(s) = 1 | + 0.706·3-s − 4.40·5-s − 7-s − 2.50·9-s + 1.09·11-s − 2·13-s − 3.11·15-s + 17-s − 0.706·21-s − 1.41·23-s + 14.4·25-s − 3.88·27-s + 5.32·29-s + 10.1·31-s + 0.772·33-s + 4.40·35-s − 3.09·37-s − 1.41·39-s + 11.2·41-s − 3.21·43-s + 11.0·45-s + 11.0·47-s + 49-s + 0.706·51-s + 6.31·53-s − 4.81·55-s − 2.18·59-s + ⋯ |
| L(s) = 1 | + 0.407·3-s − 1.97·5-s − 0.377·7-s − 0.833·9-s + 0.329·11-s − 0.554·13-s − 0.804·15-s + 0.242·17-s − 0.154·21-s − 0.294·23-s + 2.88·25-s − 0.748·27-s + 0.988·29-s + 1.81·31-s + 0.134·33-s + 0.745·35-s − 0.508·37-s − 0.226·39-s + 1.74·41-s − 0.489·43-s + 1.64·45-s + 1.61·47-s + 0.142·49-s + 0.0989·51-s + 0.867·53-s − 0.649·55-s − 0.284·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 0.706T + 3T^{2} \) |
| 5 | \( 1 + 4.40T + 5T^{2} \) |
| 11 | \( 1 - 1.09T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 + 1.29T + 61T^{2} \) |
| 67 | \( 1 - 2.08T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.57112528694490540796941048450, −7.07262009747282027224203392649, −6.25709776062753023637110105062, −5.36209948564928891416979417902, −4.30464540395139742565997541484, −4.09096277293632259606783324496, −2.94867858409933930547363617605, −2.76583735985730193786449659680, −0.986516202633092642417065190050, 0,
0.986516202633092642417065190050, 2.76583735985730193786449659680, 2.94867858409933930547363617605, 4.09096277293632259606783324496, 4.30464540395139742565997541484, 5.36209948564928891416979417902, 6.25709776062753023637110105062, 7.07262009747282027224203392649, 7.57112528694490540796941048450
