| L(s) = 1 | − 2·7-s − 4·11-s + 13-s − 4·17-s + 2·19-s + 8·23-s − 2·31-s + 10·37-s + 10·41-s − 3·49-s − 8·53-s + 4·59-s + 2·61-s − 2·67-s − 2·73-s + 8·77-s + 12·83-s + 6·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s + 0.458·19-s + 1.66·23-s − 0.359·31-s + 1.64·37-s + 1.56·41-s − 3/7·49-s − 1.09·53-s + 0.520·59-s + 0.256·61-s − 0.244·67-s − 0.234·73-s + 0.911·77-s + 1.31·83-s + 0.635·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.669204178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.669204178\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.57878336516048, −13.24619278054160, −12.87834492806575, −12.66921879708226, −11.76957863029562, −11.28945613051895, −10.84019899314324, −10.50326096320183, −9.771362323137270, −9.306099284556949, −9.006123663852269, −8.285174670741109, −7.687325432934008, −7.364012697023383, −6.636110450079199, −6.225662299748320, −5.656878412769937, −4.968441740424268, −4.619934903400670, −3.824733740898902, −3.169527120459890, −2.694298613196042, −2.170774701064016, −1.124294336541760, −0.4496115012329545,
0.4496115012329545, 1.124294336541760, 2.170774701064016, 2.694298613196042, 3.169527120459890, 3.824733740898902, 4.619934903400670, 4.968441740424268, 5.656878412769937, 6.225662299748320, 6.636110450079199, 7.364012697023383, 7.687325432934008, 8.285174670741109, 9.006123663852269, 9.306099284556949, 9.771362323137270, 10.50326096320183, 10.84019899314324, 11.28945613051895, 11.76957863029562, 12.66921879708226, 12.87834492806575, 13.24619278054160, 13.57878336516048
