L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s + 4·11-s − 2·12-s + 16-s − 6·17-s − 18-s + 6·19-s − 4·22-s − 23-s + 2·24-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s − 32-s − 8·33-s + 6·34-s + 36-s − 2·37-s − 6·38-s + 10·41-s − 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s − 0.852·22-s − 0.208·23-s + 0.408·24-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 1.39·33-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.973·38-s + 1.56·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7308049495\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7308049495\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.134247104139115818522316609574, −8.344509010663835111404645714273, −7.39667028884814206023479072290, −6.52018681868209304657633621308, −6.21202941767062653155129486942, −5.16991568706649202295744768194, −4.32810515504939812912241814943, −3.15350652511232558418559400816, −1.79791321086234584024707488474, −0.66273022074445771136649087560,
0.66273022074445771136649087560, 1.79791321086234584024707488474, 3.15350652511232558418559400816, 4.32810515504939812912241814943, 5.16991568706649202295744768194, 6.21202941767062653155129486942, 6.52018681868209304657633621308, 7.39667028884814206023479072290, 8.344509010663835111404645714273, 9.134247104139115818522316609574