| L(s) = 1 | − 7-s + 11-s + 2·13-s − 4·17-s + 2·19-s − 5·25-s + 2·29-s + 10·31-s − 10·37-s + 4·41-s − 8·43-s − 6·47-s + 49-s − 2·53-s − 4·59-s + 10·61-s − 12·67-s − 4·71-s + 4·73-s − 77-s + 14·83-s + 14·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s + 0.554·13-s − 0.970·17-s + 0.458·19-s − 25-s + 0.371·29-s + 1.79·31-s − 1.64·37-s + 0.624·41-s − 1.21·43-s − 0.875·47-s + 1/7·49-s − 0.274·53-s − 0.520·59-s + 1.28·61-s − 1.46·67-s − 0.474·71-s + 0.468·73-s − 0.113·77-s + 1.53·83-s + 1.48·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.01156592391198, −14.27755855884188, −13.85621216797873, −13.34386558264070, −13.06009124656530, −12.21778098082627, −11.74497954258622, −11.49923970681252, −10.62391781428921, −10.28346865726226, −9.684431797669250, −9.135007645382029, −8.568027688831893, −8.145051944889743, −7.414820170677361, −6.822834890663002, −6.244456780699538, −5.962240528774417, −4.880990642102002, −4.688384126685887, −3.662484323082473, −3.395018932444410, −2.476637091719445, −1.808880493389993, −0.9619986369372896, 0,
0.9619986369372896, 1.808880493389993, 2.476637091719445, 3.395018932444410, 3.662484323082473, 4.688384126685887, 4.880990642102002, 5.962240528774417, 6.244456780699538, 6.822834890663002, 7.414820170677361, 8.145051944889743, 8.568027688831893, 9.135007645382029, 9.684431797669250, 10.28346865726226, 10.62391781428921, 11.49923970681252, 11.74497954258622, 12.21778098082627, 13.06009124656530, 13.34386558264070, 13.85621216797873, 14.27755855884188, 15.01156592391198
