| L(s) = 1 | + 1.54·2-s + 2.93·3-s + 0.400·4-s − 0.454·5-s + 4.54·6-s + 3.41·7-s − 2.47·8-s + 5.61·9-s − 0.703·10-s + 1.46·11-s + 1.17·12-s − 3.08·13-s + 5.28·14-s − 1.33·15-s − 4.64·16-s − 2.82·17-s + 8.69·18-s + 2.29·19-s − 0.181·20-s + 10.0·21-s + 2.26·22-s + 0.534·23-s − 7.27·24-s − 4.79·25-s − 4.78·26-s + 7.66·27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 1.69·3-s + 0.200·4-s − 0.203·5-s + 1.85·6-s + 1.29·7-s − 0.876·8-s + 1.87·9-s − 0.222·10-s + 0.441·11-s + 0.339·12-s − 0.856·13-s + 1.41·14-s − 0.344·15-s − 1.16·16-s − 0.684·17-s + 2.04·18-s + 0.525·19-s − 0.0406·20-s + 2.18·21-s + 0.483·22-s + 0.111·23-s − 1.48·24-s − 0.958·25-s − 0.937·26-s + 1.47·27-s + 0.258·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.381915323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.381915323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 + 0.454T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 23 | \( 1 - 0.534T + 23T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 + 4.12T + 43T^{2} \) |
| 47 | \( 1 - 0.179T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 1.88T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 8.27T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 2.45T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 4.70T + 97T^{2} \) |
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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
−9.948693524892170311092352177131, −9.196813388617659480432394696822, −8.472176906533512047544473408575, −7.75882257918198549383868254870, −6.90278484837687647942897983621, −5.45278291625261874891366651975, −4.50363661666754434962382718188, −3.90945165346639574194262277113, −2.83765543950280123061958949592, −1.87568140678530105886970604329,
1.87568140678530105886970604329, 2.83765543950280123061958949592, 3.90945165346639574194262277113, 4.50363661666754434962382718188, 5.45278291625261874891366651975, 6.90278484837687647942897983621, 7.75882257918198549383868254870, 8.472176906533512047544473408575, 9.196813388617659480432394696822, 9.948693524892170311092352177131
