L(s) = 1 | + 1.61·3-s − 1.61·7-s − 0.381·9-s − 3.61·11-s + 3·13-s + 5.47·17-s + 2.61·19-s − 2.61·21-s − 0.381·23-s − 5.47·27-s − 5.70·29-s + 8.47·31-s − 5.85·33-s − 1.47·37-s + 4.85·39-s + 10.4·41-s − 10·43-s + 4.32·47-s − 4.38·49-s + 8.85·51-s − 8.47·53-s + 4.23·57-s + 5.76·59-s + 12.3·61-s + 0.618·63-s − 9.18·67-s − 0.618·69-s + ⋯ |
L(s) = 1 | + 0.934·3-s − 0.611·7-s − 0.127·9-s − 1.09·11-s + 0.832·13-s + 1.32·17-s + 0.600·19-s − 0.571·21-s − 0.0796·23-s − 1.05·27-s − 1.05·29-s + 1.52·31-s − 1.01·33-s − 0.242·37-s + 0.777·39-s + 1.63·41-s − 1.52·43-s + 0.631·47-s − 0.625·49-s + 1.23·51-s − 1.16·53-s + 0.561·57-s + 0.750·59-s + 1.57·61-s + 0.0778·63-s − 1.12·67-s − 0.0744·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.467024270\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.467024270\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + 3.61T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 0.381T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 9.18T + 67T^{2} \) |
| 71 | \( 1 - 5.09T + 71T^{2} \) |
| 73 | \( 1 - 4.38T + 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.84437021146494478098337603453, −7.17415278201526511896819691778, −6.19698957735008582343859802201, −5.68078631207035514429886570873, −4.97286331160967026204872122929, −3.89144366598108782211667731993, −3.24417697044601482884609633308, −2.83381109574461945392989700705, −1.87199111902879004505496152366, −0.69632174587222534933793980746,
0.69632174587222534933793980746, 1.87199111902879004505496152366, 2.83381109574461945392989700705, 3.24417697044601482884609633308, 3.89144366598108782211667731993, 4.97286331160967026204872122929, 5.68078631207035514429886570873, 6.19698957735008582343859802201, 7.17415278201526511896819691778, 7.84437021146494478098337603453