L(s) = 1 | − 1.49·3-s + 0.108·5-s − 3.53·7-s − 0.779·9-s + 2.32·11-s − 4.40·13-s − 0.162·15-s + 17-s + 4.21·19-s + 5.26·21-s + 0.0933·23-s − 4.98·25-s + 5.63·27-s + 3.61·29-s − 9.03·31-s − 3.47·33-s − 0.384·35-s + 11.2·37-s + 6.55·39-s + 3.59·41-s + 4.40·43-s − 0.0849·45-s + 0.732·47-s + 5.48·49-s − 1.49·51-s + 0.124·53-s + 0.253·55-s + ⋯ |
L(s) = 1 | − 0.860·3-s + 0.0486·5-s − 1.33·7-s − 0.259·9-s + 0.702·11-s − 1.22·13-s − 0.0418·15-s + 0.242·17-s + 0.966·19-s + 1.14·21-s + 0.0194·23-s − 0.997·25-s + 1.08·27-s + 0.671·29-s − 1.62·31-s − 0.604·33-s − 0.0650·35-s + 1.85·37-s + 1.05·39-s + 0.561·41-s + 0.671·43-s − 0.0126·45-s + 0.106·47-s + 0.782·49-s − 0.208·51-s + 0.0171·53-s + 0.0342·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 0.108T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 2.32T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 - 0.0933T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.59T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 - 0.732T + 47T^{2} \) |
| 53 | \( 1 - 0.124T + 53T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 - 2.00T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.35T + 73T^{2} \) |
| 79 | \( 1 + 4.31T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 3.67T + 89T^{2} \) |
| 97 | \( 1 - 1.83T + 97T^{2} \) |
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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
−7.30684401064340112115422581573, −6.74775198767552137436448519294, −5.95587612317008294834317671281, −5.65806691754414531742039362773, −4.76067325442458044775705000683, −3.90726097281330647972964978086, −3.10044993081350354298657488299, −2.34952244148436909527931349743, −0.944240346132411559242052202911, 0,
0.944240346132411559242052202911, 2.34952244148436909527931349743, 3.10044993081350354298657488299, 3.90726097281330647972964978086, 4.76067325442458044775705000683, 5.65806691754414531742039362773, 5.95587612317008294834317671281, 6.74775198767552137436448519294, 7.30684401064340112115422581573