L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·10-s − 5·11-s − 13-s − 4·16-s + 17-s + 4·19-s − 2·20-s + 10·22-s − 9·23-s + 25-s + 2·26-s + 2·29-s + 6·31-s + 8·32-s − 2·34-s − 37-s − 8·38-s + 2·41-s + 4·43-s − 10·44-s + 18·46-s − 2·47-s − 7·49-s − 2·50-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.632·10-s − 1.50·11-s − 0.277·13-s − 16-s + 0.242·17-s + 0.917·19-s − 0.447·20-s + 2.13·22-s − 1.87·23-s + 1/5·25-s + 0.392·26-s + 0.371·29-s + 1.07·31-s + 1.41·32-s − 0.342·34-s − 0.164·37-s − 1.29·38-s + 0.312·41-s + 0.609·43-s − 1.50·44-s + 2.65·46-s − 0.291·47-s − 49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4918683727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4918683727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.374099017376586694494242566191, −8.310675389650670032157918278167, −7.928231175429140627516747972883, −7.39892219893521303470259932486, −6.34213014325481372843017070237, −5.26818854446273948734612471092, −4.36686574375873109383866958826, −3.03894414233472414118185993951, −2.01948247015713489171184655990, −0.57811857535893449186788092921,
0.57811857535893449186788092921, 2.01948247015713489171184655990, 3.03894414233472414118185993951, 4.36686574375873109383866958826, 5.26818854446273948734612471092, 6.34213014325481372843017070237, 7.39892219893521303470259932486, 7.928231175429140627516747972883, 8.310675389650670032157918278167, 9.374099017376586694494242566191