| L(s) = 1 | − 3-s − 3·5-s − 2·7-s − 2·9-s + 11-s + 4·13-s + 3·15-s − 2·17-s + 19-s + 2·21-s + 7·23-s + 4·25-s + 5·27-s + 4·29-s + 3·31-s − 33-s + 6·35-s − 5·37-s − 4·39-s + 4·41-s + 2·43-s + 6·45-s − 3·49-s + 2·51-s − 14·53-s − 3·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 0.774·15-s − 0.485·17-s + 0.229·19-s + 0.436·21-s + 1.45·23-s + 4/5·25-s + 0.962·27-s + 0.742·29-s + 0.538·31-s − 0.174·33-s + 1.01·35-s − 0.821·37-s − 0.640·39-s + 0.624·41-s + 0.304·43-s + 0.894·45-s − 3/7·49-s + 0.280·51-s − 1.92·53-s − 0.404·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.364712671397940386267305620929, −7.46289574285638064622832987068, −6.62457687766450588033437106580, −6.19200363325973212765431531729, −5.14897518396893037382365191788, −4.35505353079951103255944786086, −3.45719814183933509421212816628, −2.89416069527602817782500099230, −1.08728703583931335463487320962, 0,
1.08728703583931335463487320962, 2.89416069527602817782500099230, 3.45719814183933509421212816628, 4.35505353079951103255944786086, 5.14897518396893037382365191788, 6.19200363325973212765431531729, 6.62457687766450588033437106580, 7.46289574285638064622832987068, 8.364712671397940386267305620929
