L(s) = 1 | + 3-s − 2·5-s + 3·7-s + 9-s + 3·11-s + 13-s − 2·15-s − 5·17-s − 2·19-s + 3·21-s − 23-s − 25-s + 27-s − 29-s − 6·31-s + 3·33-s − 6·35-s − 10·37-s + 39-s − 6·41-s − 2·43-s − 2·45-s + 9·47-s + 2·49-s − 5·51-s + 8·53-s − 6·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.516·15-s − 1.21·17-s − 0.458·19-s + 0.654·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.185·29-s − 1.07·31-s + 0.522·33-s − 1.01·35-s − 1.64·37-s + 0.160·39-s − 0.937·41-s − 0.304·43-s − 0.298·45-s + 1.31·47-s + 2/7·49-s − 0.700·51-s + 1.09·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.47376661209744688599823764808, −7.06381783340297823800938500180, −6.19894826759263993526834936467, −5.26264953130168396143151821353, −4.37850430908944303534064991841, −4.03792923326725960409208458044, −3.25133838314886342057155131690, −2.06218189842336892385604536705, −1.49304463151357367156570037646, 0,
1.49304463151357367156570037646, 2.06218189842336892385604536705, 3.25133838314886342057155131690, 4.03792923326725960409208458044, 4.37850430908944303534064991841, 5.26264953130168396143151821353, 6.19894826759263993526834936467, 7.06381783340297823800938500180, 7.47376661209744688599823764808