| L(s) = 1 | + 1.34·5-s − 1.22·7-s − 4.59·11-s + 3.59·13-s + 0.305·17-s − 19-s − 4.65·23-s − 3.18·25-s + 1.65·29-s + 7.04·31-s − 1.65·35-s + 0.184·37-s + 3.43·41-s + 11.2·43-s + 5.58·47-s − 5.49·49-s − 2.94·53-s − 6.19·55-s − 11.9·59-s − 2.40·61-s + 4.84·65-s − 7.63·67-s + 1.29·71-s − 12.2·73-s + 5.63·77-s + 12.6·79-s + 8.27·83-s + ⋯ |
| L(s) = 1 | + 0.602·5-s − 0.463·7-s − 1.38·11-s + 0.997·13-s + 0.0740·17-s − 0.229·19-s − 0.970·23-s − 0.636·25-s + 0.306·29-s + 1.26·31-s − 0.279·35-s + 0.0303·37-s + 0.535·41-s + 1.71·43-s + 0.814·47-s − 0.785·49-s − 0.404·53-s − 0.835·55-s − 1.55·59-s − 0.307·61-s + 0.600·65-s − 0.933·67-s + 0.153·71-s − 1.43·73-s + 0.642·77-s + 1.42·79-s + 0.908·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.22T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 - 0.305T + 17T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 0.184T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.58T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 + 7.63T + 67T^{2} \) |
| 71 | \( 1 - 1.29T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−7.75615762116545375547692774859, −6.57071320277786755103913812594, −6.04384386472354329463679624898, −5.59522147339780347550381957293, −4.64248166393782481321817871942, −3.91762542311779338371830761667, −2.92658956904437215986219027761, −2.35535721282009452155712607934, −1.28243161944204448718105502690, 0,
1.28243161944204448718105502690, 2.35535721282009452155712607934, 2.92658956904437215986219027761, 3.91762542311779338371830761667, 4.64248166393782481321817871942, 5.59522147339780347550381957293, 6.04384386472354329463679624898, 6.57071320277786755103913812594, 7.75615762116545375547692774859
