L(s) = 1 | − 3-s + 9-s − 6·11-s − 4·13-s − 3·17-s + 4·19-s − 3·23-s − 27-s − 6·29-s − 5·31-s + 6·33-s + 8·37-s + 4·39-s − 3·41-s − 8·43-s − 9·47-s + 3·51-s + 12·53-s − 4·57-s − 6·59-s + 2·61-s − 8·67-s + 3·69-s + 9·71-s + 14·73-s + 7·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 0.727·17-s + 0.917·19-s − 0.625·23-s − 0.192·27-s − 1.11·29-s − 0.898·31-s + 1.04·33-s + 1.31·37-s + 0.640·39-s − 0.468·41-s − 1.21·43-s − 1.31·47-s + 0.420·51-s + 1.64·53-s − 0.529·57-s − 0.781·59-s + 0.256·61-s − 0.977·67-s + 0.361·69-s + 1.06·71-s + 1.63·73-s + 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.85689583962087, −13.95373781934828, −13.46312656490913, −12.99354364103986, −12.70906150287038, −11.94982251559791, −11.59403305645996, −10.99985788442091, −10.55502617992666, −9.963129315657018, −9.643896357991502, −9.018712468207479, −8.176845957883211, −7.759919129400068, −7.347842087077889, −6.776015200071014, −6.019266530373424, −5.504588571662361, −4.939947630427808, −4.712586641233871, −3.700844525485833, −3.134438779688207, −2.235245501933279, −1.969996063314251, −0.6544526216058842, 0,
0.6544526216058842, 1.969996063314251, 2.235245501933279, 3.134438779688207, 3.700844525485833, 4.712586641233871, 4.939947630427808, 5.504588571662361, 6.019266530373424, 6.776015200071014, 7.347842087077889, 7.759919129400068, 8.176845957883211, 9.018712468207479, 9.643896357991502, 9.963129315657018, 10.55502617992666, 10.99985788442091, 11.59403305645996, 11.94982251559791, 12.70906150287038, 12.99354364103986, 13.46312656490913, 13.95373781934828, 14.85689583962087