L(s) = 1 | + 1.56·2-s + 0.438·4-s − 7-s − 2.43·8-s − 11-s + 4.56·13-s − 1.56·14-s − 4.68·16-s − 5.56·17-s + 3·19-s − 1.56·22-s − 2.43·23-s + 7.12·26-s − 0.438·28-s − 3.12·29-s − 7.68·31-s − 2.43·32-s − 8.68·34-s − 9.80·37-s + 4.68·38-s + 4.68·41-s + 7.68·43-s − 0.438·44-s − 3.80·46-s − 9.56·47-s − 6·49-s + 1.99·52-s + ⋯ |
L(s) = 1 | + 1.10·2-s + 0.219·4-s − 0.377·7-s − 0.862·8-s − 0.301·11-s + 1.26·13-s − 0.417·14-s − 1.17·16-s − 1.34·17-s + 0.688·19-s − 0.332·22-s − 0.508·23-s + 1.39·26-s − 0.0828·28-s − 0.579·29-s − 1.38·31-s − 0.431·32-s − 1.48·34-s − 1.61·37-s + 0.759·38-s + 0.731·41-s + 1.17·43-s − 0.0660·44-s − 0.561·46-s − 1.39·47-s − 0.857·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 2.43T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 - 4.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 + 5.68T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.246T + 73T^{2} \) |
| 79 | \( 1 + 3.31T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 - 6.12T + 97T^{2} \) |
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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
−8.759612895202747083047485626576, −7.66106143560188642040222547039, −6.72336901131260498693715762135, −6.04382758867040788462657936823, −5.40069299310611174627209783311, −4.48942440616498401021402665048, −3.72020882184908231400452384658, −3.05640639218287422571430782294, −1.82103591568195856264319831955, 0,
1.82103591568195856264319831955, 3.05640639218287422571430782294, 3.72020882184908231400452384658, 4.48942440616498401021402665048, 5.40069299310611174627209783311, 6.04382758867040788462657936823, 6.72336901131260498693715762135, 7.66106143560188642040222547039, 8.759612895202747083047485626576