L(s) = 1 | − 2.32·2-s + 3-s + 3.39·4-s + 1.73·5-s − 2.32·6-s − 7-s − 3.22·8-s + 9-s − 4.03·10-s + 4.22·11-s + 3.39·12-s + 0.457·13-s + 2.32·14-s + 1.73·15-s + 0.716·16-s − 7.83·17-s − 2.32·18-s + 5.51·19-s + 5.89·20-s − 21-s − 9.80·22-s − 1.27·23-s − 3.22·24-s − 1.97·25-s − 1.06·26-s + 27-s − 3.39·28-s + ⋯ |
L(s) = 1 | − 1.64·2-s + 0.577·3-s + 1.69·4-s + 0.777·5-s − 0.947·6-s − 0.377·7-s − 1.14·8-s + 0.333·9-s − 1.27·10-s + 1.27·11-s + 0.978·12-s + 0.126·13-s + 0.620·14-s + 0.448·15-s + 0.179·16-s − 1.89·17-s − 0.547·18-s + 1.26·19-s + 1.31·20-s − 0.218·21-s − 2.09·22-s − 0.265·23-s − 0.659·24-s − 0.395·25-s − 0.208·26-s + 0.192·27-s − 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261665103\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261665103\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
good | 2 | \( 1 + 2.32T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 - 0.457T + 13T^{2} \) |
| 17 | \( 1 + 7.83T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 + 3.47T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 9.72T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 5.02T + 53T^{2} \) |
| 59 | \( 1 - 8.18T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 - 6.60T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 0.738T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
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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
−8.840841030047948178152666135568, −7.926131527709920133250807900903, −7.00095915346307347874533296578, −6.68073330448483660152419464055, −5.85911099940280088656921831247, −4.58886767527978354560720684388, −3.60493657760467055378609416535, −2.46496760885504086197350241653, −1.82266783925318096960766586476, −0.821671387802816714286149828896,
0.821671387802816714286149828896, 1.82266783925318096960766586476, 2.46496760885504086197350241653, 3.60493657760467055378609416535, 4.58886767527978354560720684388, 5.85911099940280088656921831247, 6.68073330448483660152419464055, 7.00095915346307347874533296578, 7.926131527709920133250807900903, 8.840841030047948178152666135568