| L(s) = 1 | − 4·7-s − 4·11-s + 6·13-s + 2·17-s + 4·19-s + 10·29-s + 4·31-s − 10·37-s − 2·41-s + 4·43-s − 8·47-s + 9·49-s − 2·53-s − 12·59-s + 10·61-s − 12·67-s − 10·73-s + 16·77-s + 4·79-s + 4·83-s + 6·89-s − 24·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 1.85·29-s + 0.718·31-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s + 1.28·61-s − 1.46·67-s − 1.17·73-s + 1.82·77-s + 0.450·79-s + 0.439·83-s + 0.635·89-s − 2.51·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.529208041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.529208041\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.16772721683583, −15.71325119683406, −15.31277365511338, −14.22716834789009, −13.77821880996657, −13.29298460315862, −12.90268332912593, −12.16029168995681, −11.75719886283108, −10.82043546518578, −10.30997131799555, −10.02652029851156, −9.190123267784658, −8.643764418259445, −8.022671535721797, −7.365570059099450, −6.487142720872989, −6.249220151101373, −5.454422971820345, −4.803725686605519, −3.790997775715735, −3.139169321814783, −2.832866758616042, −1.505772572463631, −0.5597348109658727,
0.5597348109658727, 1.505772572463631, 2.832866758616042, 3.139169321814783, 3.790997775715735, 4.803725686605519, 5.454422971820345, 6.249220151101373, 6.487142720872989, 7.365570059099450, 8.022671535721797, 8.643764418259445, 9.190123267784658, 10.02652029851156, 10.30997131799555, 10.82043546518578, 11.75719886283108, 12.16029168995681, 12.90268332912593, 13.29298460315862, 13.77821880996657, 14.22716834789009, 15.31277365511338, 15.71325119683406, 16.16772721683583
