L(s) = 1 | − 2-s + 4-s + 5-s − 2·7-s − 8-s − 10-s − 4·11-s + 13-s + 2·14-s + 16-s − 4·19-s + 20-s + 4·22-s + 25-s − 26-s − 2·28-s − 6·29-s + 4·31-s − 32-s − 2·35-s + 4·38-s − 40-s + 6·41-s − 8·43-s − 4·44-s − 3·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.338·35-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s − 3/7·49-s − 0.141·50-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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.02027993311508, −12.64300221322396, −12.22830921426505, −11.40510867506254, −11.21189309027957, −10.64822449054995, −10.18439117863116, −9.926778172766385, −9.471744979648387, −8.848960019686848, −8.591519481596928, −7.990795614981686, −7.557150495311250, −7.111633006482710, −6.444663769169107, −6.192309419462355, −5.669833645898257, −5.148437685299705, −4.555903596869559, −3.922138642948028, −3.280044487693931, −2.775958666160608, −2.345081893071376, −1.707148015882158, −1.089984051943537, 0, 0,
1.089984051943537, 1.707148015882158, 2.345081893071376, 2.775958666160608, 3.280044487693931, 3.922138642948028, 4.555903596869559, 5.148437685299705, 5.669833645898257, 6.192309419462355, 6.444663769169107, 7.111633006482710, 7.557150495311250, 7.990795614981686, 8.591519481596928, 8.848960019686848, 9.471744979648387, 9.926778172766385, 10.18439117863116, 10.64822449054995, 11.21189309027957, 11.40510867506254, 12.22830921426505, 12.64300221322396, 13.02027993311508