| L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s + 3·19-s + 4·21-s + 4·23-s − 27-s + 29-s − 8·31-s − 37-s + 39-s + 41-s + 6·43-s + 11·47-s + 9·49-s − 3·53-s − 3·57-s − 10·59-s − 4·61-s − 4·63-s + 13·67-s − 4·69-s + 9·71-s − 3·79-s + 81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.688·19-s + 0.872·21-s + 0.834·23-s − 0.192·27-s + 0.185·29-s − 1.43·31-s − 0.164·37-s + 0.160·39-s + 0.156·41-s + 0.914·43-s + 1.60·47-s + 9/7·49-s − 0.412·53-s − 0.397·57-s − 1.30·59-s − 0.512·61-s − 0.503·63-s + 1.58·67-s − 0.481·69-s + 1.06·71-s − 0.337·79-s + 1/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.114535169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114535169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.15469020632573, −13.77196053213598, −12.98422070352300, −12.86071519323135, −12.26212881210631, −11.91228890452206, −11.06375994657626, −10.81377393945164, −10.18531332224687, −9.662067501990419, −9.138960915196158, −8.949425522230653, −7.834939321545447, −7.439020817381250, −6.879138659520360, −6.417189617074209, −5.836452109680877, −5.349373877765472, −4.751973807422505, −3.926096519099231, −3.464299538560782, −2.831106113187792, −2.145187380053560, −1.114143992479107, −0.4176011193059757,
0.4176011193059757, 1.114143992479107, 2.145187380053560, 2.831106113187792, 3.464299538560782, 3.926096519099231, 4.751973807422505, 5.349373877765472, 5.836452109680877, 6.417189617074209, 6.879138659520360, 7.439020817381250, 7.834939321545447, 8.949425522230653, 9.138960915196158, 9.662067501990419, 10.18531332224687, 10.81377393945164, 11.06375994657626, 11.91228890452206, 12.26212881210631, 12.86071519323135, 12.98422070352300, 13.77196053213598, 14.15469020632573
