| L(s) = 1 | − 1.15·2-s − 2.52·3-s − 0.659·4-s − 3.10·5-s + 2.92·6-s + 3.07·8-s + 3.38·9-s + 3.59·10-s + 2.14·11-s + 1.66·12-s − 0.468·13-s + 7.84·15-s − 2.24·16-s + 3.75·17-s − 3.92·18-s − 6.67·19-s + 2.04·20-s − 2.47·22-s − 2.72·23-s − 7.78·24-s + 4.64·25-s + 0.542·26-s − 0.973·27-s + 1.21·29-s − 9.08·30-s − 6.04·31-s − 3.55·32-s + ⋯ |
| L(s) = 1 | − 0.818·2-s − 1.45·3-s − 0.329·4-s − 1.38·5-s + 1.19·6-s + 1.08·8-s + 1.12·9-s + 1.13·10-s + 0.645·11-s + 0.480·12-s − 0.129·13-s + 2.02·15-s − 0.561·16-s + 0.910·17-s − 0.923·18-s − 1.53·19-s + 0.457·20-s − 0.528·22-s − 0.568·23-s − 1.58·24-s + 0.928·25-s + 0.106·26-s − 0.187·27-s + 0.224·29-s − 1.65·30-s − 1.08·31-s − 0.628·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.15T + 2T^{2} \) |
| 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 11 | \( 1 - 2.14T + 11T^{2} \) |
| 13 | \( 1 + 0.468T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 + 6.67T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 1.21T + 29T^{2} \) |
| 31 | \( 1 + 6.04T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 - 5.33T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.76T + 97T^{2} \) |
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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
−7.70027634645228021304859722528, −7.07493482319108733908001543063, −6.43852745228609420513535509082, −5.50242589520915116071817326548, −4.83176360450776244369042157153, −4.09303683203172582955092748714, −3.58936011984590474018525486756, −1.83601575715037926856325140555, −0.73241158186483281048950257282, 0,
0.73241158186483281048950257282, 1.83601575715037926856325140555, 3.58936011984590474018525486756, 4.09303683203172582955092748714, 4.83176360450776244369042157153, 5.50242589520915116071817326548, 6.43852745228609420513535509082, 7.07493482319108733908001543063, 7.70027634645228021304859722528
