L(s) = 1 | − 2.45·2-s + 4.00·4-s − 3.55·5-s − 0.769·7-s − 4.91·8-s + 8.71·10-s − 5.40·11-s − 2.95·13-s + 1.88·14-s + 4.03·16-s + 8.13·17-s − 2.72·19-s − 14.2·20-s + 13.2·22-s − 6.87·23-s + 7.63·25-s + 7.23·26-s − 3.08·28-s + 1.15·29-s + 10.1·31-s − 0.0518·32-s − 19.9·34-s + 2.73·35-s + 8.27·37-s + 6.66·38-s + 17.4·40-s − 3.54·41-s + ⋯ |
L(s) = 1 | − 1.73·2-s + 2.00·4-s − 1.58·5-s − 0.290·7-s − 1.73·8-s + 2.75·10-s − 1.62·11-s − 0.818·13-s + 0.503·14-s + 1.00·16-s + 1.97·17-s − 0.624·19-s − 3.18·20-s + 2.82·22-s − 1.43·23-s + 1.52·25-s + 1.41·26-s − 0.582·28-s + 0.214·29-s + 1.81·31-s − 0.00916·32-s − 3.41·34-s + 0.462·35-s + 1.35·37-s + 1.08·38-s + 2.76·40-s − 0.552·41-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 5 | \( 1 + 3.55T + 5T^{2} \) |
| 7 | \( 1 + 0.769T + 7T^{2} \) |
| 11 | \( 1 + 5.40T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 8.13T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 + 6.87T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 - 9.19T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−8.036992969951948604468422497958, −7.78521068643592797232827224492, −7.10830836788694609272710033283, −6.17494858673151736770314706864, −5.13241105526716674139280438927, −4.13866433804854409898119651324, −3.05272133372443925085471644145, −2.38756115206285130549927546290, −0.844629369497922400354295152258, 0,
0.844629369497922400354295152258, 2.38756115206285130549927546290, 3.05272133372443925085471644145, 4.13866433804854409898119651324, 5.13241105526716674139280438927, 6.17494858673151736770314706864, 7.10830836788694609272710033283, 7.78521068643592797232827224492, 8.036992969951948604468422497958