L(s) = 1 | + 0.305·2-s − 1.36·3-s − 1.90·4-s + 3.46·5-s − 0.417·6-s − 1.45·7-s − 1.19·8-s − 1.13·9-s + 1.06·10-s − 1.82·11-s + 2.60·12-s − 6.40·13-s − 0.445·14-s − 4.72·15-s + 3.44·16-s + 3.18·17-s − 0.348·18-s + 3.34·19-s − 6.60·20-s + 1.98·21-s − 0.558·22-s + 5.41·23-s + 1.63·24-s + 7.01·25-s − 1.96·26-s + 5.64·27-s + 2.77·28-s + ⋯ |
L(s) = 1 | + 0.216·2-s − 0.787·3-s − 0.953·4-s + 1.55·5-s − 0.170·6-s − 0.550·7-s − 0.422·8-s − 0.379·9-s + 0.335·10-s − 0.550·11-s + 0.750·12-s − 1.77·13-s − 0.119·14-s − 1.22·15-s + 0.861·16-s + 0.771·17-s − 0.0820·18-s + 0.767·19-s − 1.47·20-s + 0.433·21-s − 0.119·22-s + 1.12·23-s + 0.332·24-s + 1.40·25-s − 0.384·26-s + 1.08·27-s + 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.085577252\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.085577252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 - 0.305T + 2T^{2} \) |
| 3 | \( 1 + 1.36T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.40T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 + 0.339T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 - 6.67T + 67T^{2} \) |
| 71 | \( 1 + 8.26T + 71T^{2} \) |
| 73 | \( 1 - 8.99T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.11T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 3.28T + 97T^{2} \) |
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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
−9.547742866806149315613026720386, −8.873849197353508666423993229509, −7.67067225053065712518072422935, −6.74933905763283823401395649377, −5.75574453176568988939922028874, −5.32078962164261456362524058709, −4.86297041843553586213497445594, −3.27748897891394692335357767940, −2.40198488623230958916521455176, −0.71501326309398181769609833026,
0.71501326309398181769609833026, 2.40198488623230958916521455176, 3.27748897891394692335357767940, 4.86297041843553586213497445594, 5.32078962164261456362524058709, 5.75574453176568988939922028874, 6.74933905763283823401395649377, 7.67067225053065712518072422935, 8.873849197353508666423993229509, 9.547742866806149315613026720386