L(s) = 1 | − 2-s − 2.33·3-s + 4-s − 3.95·5-s + 2.33·6-s − 0.471·7-s − 8-s + 2.43·9-s + 3.95·10-s − 11-s − 2.33·12-s − 0.121·13-s + 0.471·14-s + 9.21·15-s + 16-s + 0.909·17-s − 2.43·18-s − 3.95·20-s + 1.09·21-s + 22-s + 2.24·23-s + 2.33·24-s + 10.6·25-s + 0.121·26-s + 1.30·27-s − 0.471·28-s + 4.53·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.34·3-s + 0.5·4-s − 1.76·5-s + 0.952·6-s − 0.178·7-s − 0.353·8-s + 0.812·9-s + 1.24·10-s − 0.301·11-s − 0.673·12-s − 0.0338·13-s + 0.125·14-s + 2.37·15-s + 0.250·16-s + 0.220·17-s − 0.574·18-s − 0.883·20-s + 0.239·21-s + 0.213·22-s + 0.467·23-s + 0.476·24-s + 2.12·25-s + 0.0239·26-s + 0.252·27-s − 0.0890·28-s + 0.842·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2625390762\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2625390762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 5 | \( 1 + 3.95T + 5T^{2} \) |
| 7 | \( 1 + 0.471T + 7T^{2} \) |
| 13 | \( 1 + 0.121T + 13T^{2} \) |
| 17 | \( 1 - 0.909T + 17T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - 4.53T + 29T^{2} \) |
| 31 | \( 1 + 3.59T + 31T^{2} \) |
| 37 | \( 1 + 4.43T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 + 4.68T + 61T^{2} \) |
| 67 | \( 1 + 3.11T + 67T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 6.36T + 79T^{2} \) |
| 83 | \( 1 + 2.37T + 83T^{2} \) |
| 89 | \( 1 + 3.88T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.59090175214244569929423358969, −7.35066138795815524239891152324, −6.56380387540311652461908547667, −5.88579390902780443599714514877, −5.02060018968083138848792527956, −4.42640082456257111370793679830, −3.53643672856463103871853552533, −2.74497306445632990296385862578, −1.21241357458060606731142985918, −0.34910515548840773384424325709,
0.34910515548840773384424325709, 1.21241357458060606731142985918, 2.74497306445632990296385862578, 3.53643672856463103871853552533, 4.42640082456257111370793679830, 5.02060018968083138848792527956, 5.88579390902780443599714514877, 6.56380387540311652461908547667, 7.35066138795815524239891152324, 7.59090175214244569929423358969