L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s − 6·13-s + 14-s + 16-s + 2·17-s − 4·19-s − 20-s + 22-s + 4·23-s + 25-s − 6·26-s + 28-s − 6·29-s + 32-s + 2·34-s − 35-s − 2·37-s − 4·38-s − 40-s + 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 0.213·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.169·35-s − 0.328·37-s − 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.45780364847670372214692851146, −7.00335198794961680400141822957, −6.14060365313477106791111576479, −5.31650297340103739861010396929, −4.72258266822774535605816515667, −4.11607768525043534978572155650, −3.20587294887114899385867883565, −2.43674449138836173509430409765, −1.48982467409762643870191702376, 0,
1.48982467409762643870191702376, 2.43674449138836173509430409765, 3.20587294887114899385867883565, 4.11607768525043534978572155650, 4.72258266822774535605816515667, 5.31650297340103739861010396929, 6.14060365313477106791111576479, 7.00335198794961680400141822957, 7.45780364847670372214692851146