L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 4·13-s − 14-s + 16-s − 7·17-s + 7·19-s + 20-s + 25-s + 4·26-s − 28-s + 2·29-s + 32-s − 7·34-s − 35-s − 6·37-s + 7·38-s + 40-s − 10·41-s − 43-s + 4·47-s + 49-s + 50-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 + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.69·17-s + 1.60·19-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.188·28-s + 0.371·29-s + 0.176·32-s − 1.20·34-s − 0.169·35-s − 0.986·37-s + 1.13·38-s + 0.158·40-s − 1.56·41-s − 0.152·43-s + 0.583·47-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.03618212755996, −12.30250505192933, −11.99839717413454, −11.50169180537315, −11.04479975736022, −10.68235674533361, −10.05992653985230, −9.805984057757060, −9.037813462846906, −8.740123855320090, −8.374417423736268, −7.553214873083097, −7.144635406606650, −6.633465585911251, −6.330655139343359, −5.789261389832259, −5.187401106307003, −4.965090278999596, −4.200205869540747, −3.672337748350423, −3.347909344669658, −2.632595598744500, −2.183756630755403, −1.481738277457630, −0.9467831919214784, 0,
0.9467831919214784, 1.481738277457630, 2.183756630755403, 2.632595598744500, 3.347909344669658, 3.672337748350423, 4.200205869540747, 4.965090278999596, 5.187401106307003, 5.789261389832259, 6.330655139343359, 6.633465585911251, 7.144635406606650, 7.553214873083097, 8.374417423736268, 8.740123855320090, 9.037813462846906, 9.805984057757060, 10.05992653985230, 10.68235674533361, 11.04479975736022, 11.50169180537315, 11.99839717413454, 12.30250505192933, 13.03618212755996