| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s − 4·7-s + 8-s + 6·9-s + 3·11-s − 3·12-s + 6·13-s − 4·14-s + 16-s − 5·17-s + 6·18-s − 19-s + 12·21-s + 3·22-s − 23-s − 3·24-s + 6·26-s − 9·27-s − 4·28-s − 8·29-s − 8·31-s + 32-s − 9·33-s − 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 1.51·7-s + 0.353·8-s + 2·9-s + 0.904·11-s − 0.866·12-s + 1.66·13-s − 1.06·14-s + 1/4·16-s − 1.21·17-s + 1.41·18-s − 0.229·19-s + 2.61·21-s + 0.639·22-s − 0.208·23-s − 0.612·24-s + 1.17·26-s − 1.73·27-s − 0.755·28-s − 1.48·29-s − 1.43·31-s + 0.176·32-s − 1.56·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.600099421356864017505939410193, −8.731972355257570203373193744180, −7.06111447183488407981494017177, −6.56982938091023632295956008615, −6.06460876229463456544502191143, −5.33617155869951169508377290637, −4.07705153459629563022967376649, −3.55408239167589212650210573167, −1.62634003225036130509439070344, 0,
1.62634003225036130509439070344, 3.55408239167589212650210573167, 4.07705153459629563022967376649, 5.33617155869951169508377290637, 6.06460876229463456544502191143, 6.56982938091023632295956008615, 7.06111447183488407981494017177, 8.731972355257570203373193744180, 9.600099421356864017505939410193
