| L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 7-s − 3·8-s + 9-s − 10-s − 2·11-s + 12-s − 14-s + 15-s − 16-s − 17-s + 18-s − 5·19-s + 20-s + 21-s − 2·22-s + 23-s + 3·24-s + 25-s − 27-s + 28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.92157395861551, −12.68700969868554, −11.86946422513226, −11.77928588216629, −11.25174526572981, −10.60301139119406, −10.16095715505613, −9.902165694595682, −9.233998974587348, −8.633634209895126, −8.322867786845469, −7.964442079781096, −7.035308844764287, −6.727723344723457, −6.363448679345300, −5.687289638092093, −5.295226467957458, −4.718827976373101, −4.481234261249719, −3.876979203931023, −3.196846950479932, −3.010332086997498, −2.096191334721550, −1.450080628069451, −0.4537441734586891, 0,
0.4537441734586891, 1.450080628069451, 2.096191334721550, 3.010332086997498, 3.196846950479932, 3.876979203931023, 4.481234261249719, 4.718827976373101, 5.295226467957458, 5.687289638092093, 6.363448679345300, 6.727723344723457, 7.035308844764287, 7.964442079781096, 8.322867786845469, 8.633634209895126, 9.233998974587348, 9.902165694595682, 10.16095715505613, 10.60301139119406, 11.25174526572981, 11.77928588216629, 11.86946422513226, 12.68700969868554, 12.92157395861551
