| L(s) = 1 | − 5-s − 1.41·7-s − 11-s − 0.585·13-s + 3.41·17-s + 5.65·19-s − 2.82·23-s + 25-s − 0.828·29-s − 6.48·31-s + 1.41·35-s − 7.65·37-s + 10.4·41-s − 2.58·43-s + 2.82·47-s − 5·49-s + 7.17·53-s + 55-s − 10.4·59-s − 3.17·61-s + 0.585·65-s + 8.48·67-s − 3.17·71-s + 7.41·73-s + 1.41·77-s − 1.65·79-s + 0.242·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.534·7-s − 0.301·11-s − 0.162·13-s + 0.828·17-s + 1.29·19-s − 0.589·23-s + 0.200·25-s − 0.153·29-s − 1.16·31-s + 0.239·35-s − 1.25·37-s + 1.63·41-s − 0.394·43-s + 0.412·47-s − 0.714·49-s + 0.985·53-s + 0.134·55-s − 1.36·59-s − 0.406·61-s + 0.0726·65-s + 1.03·67-s − 0.376·71-s + 0.867·73-s + 0.161·77-s − 0.186·79-s + 0.0266·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 2.58T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 7.17T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 - 0.242T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 0.828T + 97T^{2} \) |
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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
−7.50252357103400788209391852026, −6.98523356230022514271352271110, −6.00762110476728763606786536682, −5.46358319187412817890142332373, −4.72026830371783159194584535189, −3.68981316652905853374038061753, −3.29548543864173623593429318866, −2.31016709144515529811238507229, −1.16765560849265682113233566948, 0,
1.16765560849265682113233566948, 2.31016709144515529811238507229, 3.29548543864173623593429318866, 3.68981316652905853374038061753, 4.72026830371783159194584535189, 5.46358319187412817890142332373, 6.00762110476728763606786536682, 6.98523356230022514271352271110, 7.50252357103400788209391852026
