L(s) = 1 | + 0.305·2-s − 1.90·4-s − 1.63·5-s − 3.53·7-s − 1.19·8-s − 0.499·10-s − 3.40·11-s − 1.61·13-s − 1.08·14-s + 3.44·16-s − 2.63·17-s − 2.77·19-s + 3.11·20-s − 1.04·22-s − 4.69·23-s − 2.32·25-s − 0.495·26-s + 6.73·28-s + 5.63·29-s − 6.83·31-s + 3.44·32-s − 0.804·34-s + 5.78·35-s − 4.59·37-s − 0.847·38-s + 1.95·40-s − 9.48·41-s + ⋯ |
L(s) = 1 | + 0.216·2-s − 0.953·4-s − 0.731·5-s − 1.33·7-s − 0.422·8-s − 0.158·10-s − 1.02·11-s − 0.449·13-s − 0.288·14-s + 0.862·16-s − 0.638·17-s − 0.636·19-s + 0.697·20-s − 0.222·22-s − 0.978·23-s − 0.464·25-s − 0.0970·26-s + 1.27·28-s + 1.04·29-s − 1.22·31-s + 0.608·32-s − 0.137·34-s + 0.977·35-s − 0.755·37-s − 0.137·38-s + 0.308·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01722755994\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01722755994\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
good | 2 | \( 1 - 0.305T + 2T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 6.83T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 2.81T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 8.73T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.242847633422625345134397798217, −8.021420358745771632584074395704, −6.82385341583147243223263683030, −6.33852821161174881263310099249, −5.23987923156950492320779658759, −4.74600348913906322313778143658, −3.65441539590819531464032308068, −3.35064234828388403690584305222, −2.11848018533228487272419516074, −0.06777933023000852423447037410,
0.06777933023000852423447037410, 2.11848018533228487272419516074, 3.35064234828388403690584305222, 3.65441539590819531464032308068, 4.74600348913906322313778143658, 5.23987923156950492320779658759, 6.33852821161174881263310099249, 6.82385341583147243223263683030, 8.021420358745771632584074395704, 8.242847633422625345134397798217