L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 23-s + 25-s + 27-s − 6·29-s − 4·33-s − 6·37-s + 2·39-s + 2·41-s − 12·43-s − 45-s − 7·49-s + 2·51-s + 10·53-s + 4·55-s + 12·59-s − 6·61-s − 2·65-s − 12·67-s + 69-s + 12·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s − 0.149·45-s − 49-s + 0.280·51-s + 1.37·53-s + 0.539·55-s + 1.56·59-s − 0.768·61-s − 0.248·65-s − 1.46·67-s + 0.120·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.77263665057989112507198715536, −7.32433188625610157879257916860, −6.46684426398263158009840156349, −5.49320497231391770290270481573, −4.94197576600261032216762525785, −3.89048555354541743783367736983, −3.31234221127398228158516411000, −2.45951695841766319642404329340, −1.42978791056950556652553478169, 0,
1.42978791056950556652553478169, 2.45951695841766319642404329340, 3.31234221127398228158516411000, 3.89048555354541743783367736983, 4.94197576600261032216762525785, 5.49320497231391770290270481573, 6.46684426398263158009840156349, 7.32433188625610157879257916860, 7.77263665057989112507198715536