| L(s) = 1 | − 5-s − 1.82·11-s − 6.41·13-s + 3.58·17-s + 7.65·19-s − 3.41·23-s + 25-s + 4.65·29-s − 7.41·31-s − 0.585·37-s + 3.41·41-s + 0.343·43-s + 10.8·47-s + 12.2·53-s + 1.82·55-s + 0.585·59-s − 10.8·61-s + 6.41·65-s + 3.07·67-s + 10.4·71-s − 10.8·73-s − 15.1·79-s − 8·83-s − 3.58·85-s − 16.9·89-s − 7.65·95-s + 9.72·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.551·11-s − 1.77·13-s + 0.869·17-s + 1.75·19-s − 0.711·23-s + 0.200·25-s + 0.864·29-s − 1.33·31-s − 0.0963·37-s + 0.533·41-s + 0.0523·43-s + 1.58·47-s + 1.68·53-s + 0.246·55-s + 0.0762·59-s − 1.38·61-s + 0.795·65-s + 0.375·67-s + 1.24·71-s − 1.26·73-s − 1.70·79-s − 0.878·83-s − 0.388·85-s − 1.79·89-s − 0.785·95-s + 0.987·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + 1.82T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 - 3.58T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 0.585T + 37T^{2} \) |
| 41 | \( 1 - 3.41T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 0.585T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 9.72T + 97T^{2} \) |
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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
−7.39147256907009360363995972721, −7.10379202550673042638583436948, −5.79063171936550874462078254278, −5.39548232140894837777117422221, −4.67304589246096366563352339139, −3.85243241081626781515390275987, −2.98000595675622198442627654925, −2.38412671210003606905580618141, −1.13345424901774665842978120655, 0,
1.13345424901774665842978120655, 2.38412671210003606905580618141, 2.98000595675622198442627654925, 3.85243241081626781515390275987, 4.67304589246096366563352339139, 5.39548232140894837777117422221, 5.79063171936550874462078254278, 7.10379202550673042638583436948, 7.39147256907009360363995972721
