| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s + 11-s − 12-s − 13-s − 2·14-s + 16-s − 17-s + 18-s + 8·19-s + 2·21-s + 22-s − 24-s − 26-s − 27-s − 2·28-s + 10·29-s − 6·31-s + 32-s − 33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.436·21-s + 0.213·22-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + 1.85·29-s − 1.07·31-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.62244093276389, −12.14501057172440, −12.07292803826268, −11.46702880947018, −11.09655062578503, −10.44612238779680, −10.15509094474557, −9.637997212361805, −9.220317732433614, −8.745129714825877, −7.865581470782946, −7.721273331862003, −6.934610157610312, −6.674334540850276, −6.323499538880191, −5.646912485516436, −5.229249548769413, −4.898246527623664, −4.267118891751621, −3.660172014561731, −3.284197236103798, −2.739925085039118, −2.134924469524783, −1.319104320777801, −0.8505348858658936, 0,
0.8505348858658936, 1.319104320777801, 2.134924469524783, 2.739925085039118, 3.284197236103798, 3.660172014561731, 4.267118891751621, 4.898246527623664, 5.229249548769413, 5.646912485516436, 6.323499538880191, 6.674334540850276, 6.934610157610312, 7.721273331862003, 7.865581470782946, 8.745129714825877, 9.220317732433614, 9.637997212361805, 10.15509094474557, 10.44612238779680, 11.09655062578503, 11.46702880947018, 12.07292803826268, 12.14501057172440, 12.62244093276389
