L(s) = 1 | + 4·7-s + 4·11-s − 4·13-s + 6·17-s − 4·19-s + 4·23-s + 4·29-s − 4·37-s − 8·41-s + 12·47-s + 9·49-s + 2·53-s − 12·59-s + 2·61-s − 8·67-s − 8·71-s + 16·73-s + 16·77-s − 8·79-s + 8·83-s − 16·91-s + 8·97-s + 12·101-s + 4·103-s + 8·107-s + 18·109-s + 10·113-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 1.20·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 0.742·29-s − 0.657·37-s − 1.24·41-s + 1.75·47-s + 9/7·49-s + 0.274·53-s − 1.56·59-s + 0.256·61-s − 0.977·67-s − 0.949·71-s + 1.87·73-s + 1.82·77-s − 0.900·79-s + 0.878·83-s − 1.67·91-s + 0.812·97-s + 1.19·101-s + 0.394·103-s + 0.773·107-s + 1.72·109-s + 0.940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.228756179\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228756179\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.124226427085775873087377763953, −8.530509017024547845452001076649, −7.66781471378289723267918049344, −7.06755102843315759432168916353, −6.02210466849977928945960075358, −5.03636598174767300551022659529, −4.50041935630769765716651416823, −3.38857402541799257411199641138, −2.10097114951257368394573332804, −1.12273059786013059192203656615,
1.12273059786013059192203656615, 2.10097114951257368394573332804, 3.38857402541799257411199641138, 4.50041935630769765716651416823, 5.03636598174767300551022659529, 6.02210466849977928945960075358, 7.06755102843315759432168916353, 7.66781471378289723267918049344, 8.530509017024547845452001076649, 9.124226427085775873087377763953