| L(s) = 1 | − 3-s − 2·5-s − 2·7-s − 2·9-s − 2·11-s + 7·13-s + 2·15-s − 3·17-s + 5·19-s + 2·21-s + 3·23-s − 25-s + 5·27-s − 9·29-s + 8·31-s + 2·33-s + 4·35-s + 3·37-s − 7·39-s − 2·41-s − 4·43-s + 4·45-s + 10·47-s − 3·49-s + 3·51-s + 4·55-s − 5·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s − 2/3·9-s − 0.603·11-s + 1.94·13-s + 0.516·15-s − 0.727·17-s + 1.14·19-s + 0.436·21-s + 0.625·23-s − 1/5·25-s + 0.962·27-s − 1.67·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s + 0.493·37-s − 1.12·39-s − 0.312·41-s − 0.609·43-s + 0.596·45-s + 1.45·47-s − 3/7·49-s + 0.420·51-s + 0.539·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 53 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.32872477639942, −12.92666342951186, −12.48787775241581, −11.77734803260430, −11.47448468398827, −11.14004333724308, −10.80634821339546, −10.15250331280713, −9.622613002698881, −9.019160286967072, −8.627777859313646, −8.146781052204798, −7.657618963278585, −7.092368767283701, −6.522598210127919, −6.072840989465756, −5.660814284852613, −5.142828217251920, −4.447333513409090, −3.886206295611624, −3.336553503352944, −3.029093964543137, −2.241580487119659, −1.278150415515250, −0.6663615476411986, 0,
0.6663615476411986, 1.278150415515250, 2.241580487119659, 3.029093964543137, 3.336553503352944, 3.886206295611624, 4.447333513409090, 5.142828217251920, 5.660814284852613, 6.072840989465756, 6.522598210127919, 7.092368767283701, 7.657618963278585, 8.146781052204798, 8.627777859313646, 9.019160286967072, 9.622613002698881, 10.15250331280713, 10.80634821339546, 11.14004333724308, 11.47448468398827, 11.77734803260430, 12.48787775241581, 12.92666342951186, 13.32872477639942
