| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 4·13-s + 15-s + 16-s + 6·17-s + 18-s − 8·19-s + 20-s − 2·22-s − 23-s + 24-s + 25-s + 4·26-s + 27-s + 4·29-s + 30-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.742·29-s + 0.182·30-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.015845354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015845354\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−10.46006842950388688821498725352, −9.778476790971364876210604546999, −8.514839073741154480133769062900, −8.012185614946042132726363856353, −6.73556129089751089343223996833, −5.99007031258734046981490521178, −4.96141389876974978877383998444, −3.84335009814648903472376632301, −2.88411126258645777238121403143, −1.64793340157193246414261963354,
1.64793340157193246414261963354, 2.88411126258645777238121403143, 3.84335009814648903472376632301, 4.96141389876974978877383998444, 5.99007031258734046981490521178, 6.73556129089751089343223996833, 8.012185614946042132726363856353, 8.514839073741154480133769062900, 9.778476790971364876210604546999, 10.46006842950388688821498725352
