| L(s) = 1 | + 4·7-s + 28·11-s − 16·13-s − 108·17-s + 32·19-s + 28·23-s + 238·29-s − 180·31-s − 40·37-s − 422·41-s + 276·43-s − 60·47-s − 327·49-s − 220·53-s + 804·59-s − 358·61-s − 884·67-s + 64·71-s − 152·73-s + 112·77-s − 932·79-s + 1.29e3·83-s + 1.14e3·89-s − 64·91-s + 824·97-s + 1.29e3·101-s − 1.60e3·103-s + ⋯ |
| L(s) = 1 | + 0.215·7-s + 0.767·11-s − 0.341·13-s − 1.54·17-s + 0.386·19-s + 0.253·23-s + 1.52·29-s − 1.04·31-s − 0.177·37-s − 1.60·41-s + 0.978·43-s − 0.186·47-s − 0.953·49-s − 0.570·53-s + 1.77·59-s − 0.751·61-s − 1.61·67-s + 0.106·71-s − 0.243·73-s + 0.165·77-s − 1.32·79-s + 1.70·83-s + 1.36·89-s − 0.0737·91-s + 0.862·97-s + 1.27·101-s − 1.53·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 108 T + p^{3} T^{2} \) |
| 19 | \( 1 - 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 29 | \( 1 - 238 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 + 40 T + p^{3} T^{2} \) |
| 41 | \( 1 + 422 T + p^{3} T^{2} \) |
| 43 | \( 1 - 276 T + p^{3} T^{2} \) |
| 47 | \( 1 + 60 T + p^{3} T^{2} \) |
| 53 | \( 1 + 220 T + p^{3} T^{2} \) |
| 59 | \( 1 - 804 T + p^{3} T^{2} \) |
| 61 | \( 1 + 358 T + p^{3} T^{2} \) |
| 67 | \( 1 + 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 64 T + p^{3} T^{2} \) |
| 73 | \( 1 + 152 T + p^{3} T^{2} \) |
| 79 | \( 1 + 932 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1292 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 - 824 T + p^{3} T^{2} \) |
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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
−8.702628473260225045205616616402, −7.71058337948042945776397749914, −6.84397647721981475320609664147, −6.29334487080988306169535687165, −5.11989959780106811761094906985, −4.45555966583825479884112882114, −3.46267035851582352483494085760, −2.36575905497600618025349368315, −1.33242656058482701065154316913, 0,
1.33242656058482701065154316913, 2.36575905497600618025349368315, 3.46267035851582352483494085760, 4.45555966583825479884112882114, 5.11989959780106811761094906985, 6.29334487080988306169535687165, 6.84397647721981475320609664147, 7.71058337948042945776397749914, 8.702628473260225045205616616402
