| L(s) = 1 | − 2·7-s + 2·13-s − 6·17-s + 4·19-s + 6·23-s + 6·29-s − 4·31-s + 2·37-s − 6·41-s − 10·43-s − 6·47-s − 3·49-s + 6·53-s + 12·59-s − 2·61-s + 2·67-s + 12·71-s − 2·73-s + 8·79-s − 6·83-s + 6·89-s − 4·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s + 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.244·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.635·89-s − 0.419·91-s − 0.203·97-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
| \(L(1)\) |
\(\approx\) |
\(1.660815438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660815438\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.10231799646676, −15.60594932159956, −15.08164117769177, −14.50254108097473, −13.68117025139653, −13.26877992957726, −12.97765204969365, −12.14899797607482, −11.53596431277692, −11.07451703988198, −10.41952529541142, −9.759679839577908, −9.270202559551044, −8.587397827393317, −8.170467946518908, −7.099165874972361, −6.759338460355620, −6.247319813378470, −5.267533500282371, −4.849370716053494, −3.871203817773238, −3.281743979221848, −2.580395765611843, −1.595592577591718, −0.5718175825209847,
0.5718175825209847, 1.595592577591718, 2.580395765611843, 3.281743979221848, 3.871203817773238, 4.849370716053494, 5.267533500282371, 6.247319813378470, 6.759338460355620, 7.099165874972361, 8.170467946518908, 8.587397827393317, 9.270202559551044, 9.759679839577908, 10.41952529541142, 11.07451703988198, 11.53596431277692, 12.14899797607482, 12.97765204969365, 13.26877992957726, 13.68117025139653, 14.50254108097473, 15.08164117769177, 15.60594932159956, 16.10231799646676
