L(s) = 1 | − 2.59·2-s − 1.82·3-s + 4.74·4-s + 5-s + 4.74·6-s + 7-s − 7.11·8-s + 0.333·9-s − 2.59·10-s + 1.24·11-s − 8.65·12-s + 0.0484·13-s − 2.59·14-s − 1.82·15-s + 8.98·16-s − 7.88·17-s − 0.865·18-s + 0.643·19-s + 4.74·20-s − 1.82·21-s − 3.23·22-s − 23-s + 12.9·24-s + 25-s − 0.125·26-s + 4.86·27-s + 4.74·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 1.05·3-s + 2.37·4-s + 0.447·5-s + 1.93·6-s + 0.377·7-s − 2.51·8-s + 0.111·9-s − 0.820·10-s + 0.376·11-s − 2.49·12-s + 0.0134·13-s − 0.693·14-s − 0.471·15-s + 2.24·16-s − 1.91·17-s − 0.204·18-s + 0.147·19-s + 1.05·20-s − 0.398·21-s − 0.690·22-s − 0.208·23-s + 2.65·24-s + 0.200·25-s − 0.0246·26-s + 0.936·27-s + 0.895·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 3 | \( 1 + 1.82T + 3T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 0.0484T + 13T^{2} \) |
| 17 | \( 1 + 7.88T + 17T^{2} \) |
| 19 | \( 1 - 0.643T + 19T^{2} \) |
| 29 | \( 1 - 2.42T + 29T^{2} \) |
| 31 | \( 1 + 2.85T + 31T^{2} \) |
| 37 | \( 1 - 3.11T + 37T^{2} \) |
| 41 | \( 1 - 0.326T + 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 2.14T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 0.800T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.62T + 97T^{2} \) |
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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
−9.795900127185310171089730475928, −9.009449976313259359824212729206, −8.391532866599639238100378754356, −7.27378436324199274260162841524, −6.51218944618769374661356895843, −5.87126161090660397040219221971, −4.57658828397951865004117596693, −2.60045519432394827449314395899, −1.43522884665067627145455035000, 0,
1.43522884665067627145455035000, 2.60045519432394827449314395899, 4.57658828397951865004117596693, 5.87126161090660397040219221971, 6.51218944618769374661356895843, 7.27378436324199274260162841524, 8.391532866599639238100378754356, 9.009449976313259359824212729206, 9.795900127185310171089730475928