L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 2·11-s − 4.16·13-s + 14-s + 16-s − 7.32·17-s + 3·19-s − 2·22-s − 1.16·23-s + 4.16·26-s − 28-s + 1.83·29-s − 6.32·31-s − 32-s + 7.32·34-s − 7.48·37-s − 3·38-s + 4·41-s − 3.16·43-s + 2·44-s + 1.16·46-s + 10.4·47-s + 49-s − 4.16·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.15·13-s + 0.267·14-s + 0.250·16-s − 1.77·17-s + 0.688·19-s − 0.426·22-s − 0.242·23-s + 0.816·26-s − 0.188·28-s + 0.341·29-s − 1.13·31-s − 0.176·32-s + 1.25·34-s − 1.23·37-s − 0.486·38-s + 0.624·41-s − 0.482·43-s + 0.301·44-s + 0.171·46-s + 1.52·47-s + 0.142·49-s − 0.577·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8246936616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8246936616\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 + 7.32T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.162T + 53T^{2} \) |
| 59 | \( 1 + 0.324T + 59T^{2} \) |
| 61 | \( 1 + 3.83T + 61T^{2} \) |
| 67 | \( 1 - 3.48T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4.32T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.42659311195923015221784139978, −7.25949067644908833017585668190, −6.49777134664145749342429645049, −5.81599434161416013239979454738, −4.93341165670980518877937490351, −4.19781980102160915454060471455, −3.31037654377670398631283679486, −2.42865912675474373957756922350, −1.74956366764449587655182591078, −0.46756574065012921790133985023,
0.46756574065012921790133985023, 1.74956366764449587655182591078, 2.42865912675474373957756922350, 3.31037654377670398631283679486, 4.19781980102160915454060471455, 4.93341165670980518877937490351, 5.81599434161416013239979454738, 6.49777134664145749342429645049, 7.25949067644908833017585668190, 7.42659311195923015221784139978