L(s) = 1 | − 7-s + 6·11-s − 2·13-s − 4·19-s − 6·23-s − 6·29-s + 8·31-s − 2·37-s − 12·41-s + 4·43-s + 12·47-s + 49-s − 6·53-s − 10·61-s − 8·67-s − 6·71-s + 10·73-s − 6·77-s − 4·79-s − 12·83-s − 12·89-s + 2·91-s + 10·97-s + 12·101-s − 8·103-s − 6·107-s + 14·109-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.25·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 1.87·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.28·61-s − 0.977·67-s − 0.712·71-s + 1.17·73-s − 0.683·77-s − 0.450·79-s − 1.31·83-s − 1.27·89-s + 0.209·91-s + 1.01·97-s + 1.19·101-s − 0.788·103-s − 0.580·107-s + 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.61473070158703444317930826316, −6.90118043842526761332103430613, −6.29196859144844433427154246208, −5.76300888063010329348501382463, −4.58672451802024691356050351890, −4.07678461275524983450247211516, −3.31136018034580350992724164179, −2.23103727662521916138052881106, −1.37630585382137282703817250396, 0,
1.37630585382137282703817250396, 2.23103727662521916138052881106, 3.31136018034580350992724164179, 4.07678461275524983450247211516, 4.58672451802024691356050351890, 5.76300888063010329348501382463, 6.29196859144844433427154246208, 6.90118043842526761332103430613, 7.61473070158703444317930826316