L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 13-s + 4·14-s + 16-s + 6·17-s − 4·19-s + 20-s − 6·23-s + 25-s − 26-s − 4·28-s + 6·29-s − 10·31-s − 32-s − 6·34-s − 4·35-s − 10·37-s + 4·38-s − 40-s + 6·41-s − 4·43-s + 6·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s − 1.25·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s − 1.02·34-s − 0.676·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.517510424754277050848912412444, −8.667268438616050591911554892398, −7.82435414450686800964016289834, −6.77961088113047821902650713100, −6.22432248611604168852689673877, −5.38172261903332151011080059479, −3.78815872562505355810689730454, −2.98861435930463534169178574530, −1.68435269991504536242240813005, 0,
1.68435269991504536242240813005, 2.98861435930463534169178574530, 3.78815872562505355810689730454, 5.38172261903332151011080059479, 6.22432248611604168852689673877, 6.77961088113047821902650713100, 7.82435414450686800964016289834, 8.667268438616050591911554892398, 9.517510424754277050848912412444