L(s) = 1 | + 5-s − 7-s + 11-s + 2·17-s − 6·19-s − 8·23-s + 25-s + 4·31-s − 35-s − 2·37-s − 8·41-s + 8·43-s + 12·47-s + 49-s + 10·53-s + 55-s + 12·59-s − 2·61-s − 4·67-s + 8·71-s − 12·73-s − 77-s − 10·79-s + 2·83-s + 2·85-s + 6·89-s − 6·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.301·11-s + 0.485·17-s − 1.37·19-s − 1.66·23-s + 1/5·25-s + 0.718·31-s − 0.169·35-s − 0.328·37-s − 1.24·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + 1.37·53-s + 0.134·55-s + 1.56·59-s − 0.256·61-s − 0.488·67-s + 0.949·71-s − 1.40·73-s − 0.113·77-s − 1.12·79-s + 0.219·83-s + 0.216·85-s + 0.635·89-s − 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.45104284688500, −15.97054737373478, −15.25403164423312, −14.83077852914301, −14.06598345093488, −13.75445676748822, −13.11809360371039, −12.39172141931366, −12.09709218185192, −11.39678525370189, −10.51880093256021, −10.21767046935531, −9.685230221255115, −8.821331348343823, −8.512689625826605, −7.692521844668009, −6.993634090274054, −6.340448108172127, −5.850384783389313, −5.207008190036075, −4.108295659073683, −3.917576438455564, −2.714954032501265, −2.170804326705937, −1.187004184304927, 0,
1.187004184304927, 2.170804326705937, 2.714954032501265, 3.917576438455564, 4.108295659073683, 5.207008190036075, 5.850384783389313, 6.340448108172127, 6.993634090274054, 7.692521844668009, 8.512689625826605, 8.821331348343823, 9.685230221255115, 10.21767046935531, 10.51880093256021, 11.39678525370189, 12.09709218185192, 12.39172141931366, 13.11809360371039, 13.75445676748822, 14.06598345093488, 14.83077852914301, 15.25403164423312, 15.97054737373478, 16.45104284688500