L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 13-s + 16-s − 8·19-s + 20-s − 4·23-s + 25-s − 26-s − 2·29-s + 4·31-s − 32-s − 6·37-s + 8·38-s − 40-s + 10·41-s + 4·46-s − 8·47-s − 7·49-s − 50-s + 52-s − 6·53-s + 2·58-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 0.277·13-s + 1/4·16-s − 1.83·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s − 0.196·26-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.986·37-s + 1.29·38-s − 0.158·40-s + 1.56·41-s + 0.589·46-s − 1.16·47-s − 49-s − 0.141·50-s + 0.138·52-s − 0.824·53-s + 0.262·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.02282582059241, −12.55899183353224, −12.12426263261914, −11.56875285368483, −11.06533605764447, −10.75396031765157, −10.23663330447960, −9.909678916359653, −9.349476437113264, −8.967951981397371, −8.368155090165391, −8.194342787589204, −7.565796602010353, −7.057769370996696, −6.419790663678345, −6.184436172724185, −5.803221215413282, −4.988911792682486, −4.562247419158063, −3.953292041085283, −3.435473937671563, −2.632426626947629, −2.348776118692214, −1.540840811451763, −1.310489170118160, 0, 0,
1.310489170118160, 1.540840811451763, 2.348776118692214, 2.632426626947629, 3.435473937671563, 3.953292041085283, 4.562247419158063, 4.988911792682486, 5.803221215413282, 6.184436172724185, 6.419790663678345, 7.057769370996696, 7.565796602010353, 8.194342787589204, 8.368155090165391, 8.967951981397371, 9.349476437113264, 9.909678916359653, 10.23663330447960, 10.75396031765157, 11.06533605764447, 11.56875285368483, 12.12426263261914, 12.55899183353224, 13.02282582059241