L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 2·13-s + 14-s + 16-s + 5.12·17-s − 7.12·19-s + 20-s − 22-s + 1.12·23-s + 25-s − 2·26-s − 28-s − 8.24·29-s + 1.12·31-s − 32-s − 5.12·34-s − 35-s − 1.12·37-s + 7.12·38-s − 40-s − 0.876·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 0.250·16-s + 1.24·17-s − 1.63·19-s + 0.223·20-s − 0.213·22-s + 0.234·23-s + 0.200·25-s − 0.392·26-s − 0.188·28-s − 1.53·29-s + 0.201·31-s − 0.176·32-s − 0.878·34-s − 0.169·35-s − 0.184·37-s + 1.15·38-s − 0.158·40-s − 0.136·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 0.876T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 0.876T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.77149743030275547087844509666, −6.86351525870515771953629407719, −6.28661359246940961805771077587, −5.75210513860716419283317389935, −4.81697881762992402594272515941, −3.77847458107968562534567290533, −3.11955513324193006234013032071, −2.05047783562099962339180099616, −1.31392596698359125325415104831, 0,
1.31392596698359125325415104831, 2.05047783562099962339180099616, 3.11955513324193006234013032071, 3.77847458107968562534567290533, 4.81697881762992402594272515941, 5.75210513860716419283317389935, 6.28661359246940961805771077587, 6.86351525870515771953629407719, 7.77149743030275547087844509666