| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 5·11-s − 13-s + 14-s + 16-s + 4·17-s + 2·19-s + 5·22-s − 5·23-s − 5·25-s − 26-s + 28-s − 4·29-s + 31-s + 32-s + 4·34-s + 7·37-s + 2·38-s + 9·41-s − 12·43-s + 5·44-s − 5·46-s + 7·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s + 1.06·22-s − 1.04·23-s − 25-s − 0.196·26-s + 0.188·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.685·34-s + 1.15·37-s + 0.324·38-s + 1.40·41-s − 1.82·43-s + 0.753·44-s − 0.737·46-s + 1.02·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.093420062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.093420062\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.612571086844589429454518424632, −8.469152938024698877370110317561, −7.68409626813723475327909628947, −6.90643848137944117202320139495, −5.98244664513634320441588551536, −5.36737041037488579734906212522, −4.15982302022081941279948742151, −3.71179494687009087455073957292, −2.36943591207239051047957475876, −1.23766611312278004155286222469,
1.23766611312278004155286222469, 2.36943591207239051047957475876, 3.71179494687009087455073957292, 4.15982302022081941279948742151, 5.36737041037488579734906212522, 5.98244664513634320441588551536, 6.90643848137944117202320139495, 7.68409626813723475327909628947, 8.469152938024698877370110317561, 9.612571086844589429454518424632
