L(s) = 1 | − 1.40·3-s + 3.37·5-s − 3.92·7-s − 1.02·9-s + 1.19·11-s + 1.76·13-s − 4.73·15-s − 17-s − 5.97·19-s + 5.51·21-s − 1.06·23-s + 6.38·25-s + 5.65·27-s + 5.79·29-s + 0.924·31-s − 1.67·33-s − 13.2·35-s − 6.65·37-s − 2.48·39-s + 8.89·41-s + 8.28·43-s − 3.46·45-s − 1.68·47-s + 8.40·49-s + 1.40·51-s + 3.41·53-s + 4.03·55-s + ⋯ |
L(s) = 1 | − 0.810·3-s + 1.50·5-s − 1.48·7-s − 0.342·9-s + 0.360·11-s + 0.490·13-s − 1.22·15-s − 0.242·17-s − 1.36·19-s + 1.20·21-s − 0.221·23-s + 1.27·25-s + 1.08·27-s + 1.07·29-s + 0.165·31-s − 0.292·33-s − 2.23·35-s − 1.09·37-s − 0.397·39-s + 1.38·41-s + 1.26·43-s − 0.517·45-s − 0.245·47-s + 1.20·49-s + 0.196·51-s + 0.469·53-s + 0.543·55-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 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 19 | \( 1 + 5.97T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 - 0.924T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 - 8.89T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 + 1.68T + 47T^{2} \) |
| 53 | \( 1 - 3.41T + 53T^{2} \) |
| 61 | \( 1 + 6.47T + 61T^{2} \) |
| 67 | \( 1 + 1.63T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 9.00T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.09131431582211668028095539483, −6.50135987871663337937960707342, −6.02607211859839925800754328272, −5.82424505708533525695523842691, −4.80959928495458629465273737975, −3.95270195525383041524598344729, −2.91463566852338611146686557643, −2.31689394536841048899011693760, −1.15808945461239984875275686049, 0,
1.15808945461239984875275686049, 2.31689394536841048899011693760, 2.91463566852338611146686557643, 3.95270195525383041524598344729, 4.80959928495458629465273737975, 5.82424505708533525695523842691, 6.02607211859839925800754328272, 6.50135987871663337937960707342, 7.09131431582211668028095539483