| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s + 4·5-s + 4·6-s + 9-s + 8·10-s + 3·11-s + 4·12-s − 5·13-s + 8·15-s − 4·16-s − 3·17-s + 2·18-s + 2·19-s + 8·20-s + 6·22-s − 23-s + 11·25-s − 10·26-s − 4·27-s + 6·29-s + 16·30-s − 31-s − 8·32-s + 6·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s + 1.78·5-s + 1.63·6-s + 1/3·9-s + 2.52·10-s + 0.904·11-s + 1.15·12-s − 1.38·13-s + 2.06·15-s − 16-s − 0.727·17-s + 0.471·18-s + 0.458·19-s + 1.78·20-s + 1.27·22-s − 0.208·23-s + 11/5·25-s − 1.96·26-s − 0.769·27-s + 1.11·29-s + 2.92·30-s − 0.179·31-s − 1.41·32-s + 1.04·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.652267835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.652267835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−9.351106314858612004769456814696, −8.706626615531981381764341189714, −7.46238598684507898242070539770, −6.53587314689822873386512255529, −5.98891101010360117124996175140, −5.05622923091780464116124937114, −4.38242541046099734716547640882, −3.16024004253953616612721273066, −2.55482539715432140211864034960, −1.80047517979266187156874628712,
1.80047517979266187156874628712, 2.55482539715432140211864034960, 3.16024004253953616612721273066, 4.38242541046099734716547640882, 5.05622923091780464116124937114, 5.98891101010360117124996175140, 6.53587314689822873386512255529, 7.46238598684507898242070539770, 8.706626615531981381764341189714, 9.351106314858612004769456814696
