L(s) = 1 | + 2·4-s + 5i·7-s − 5i·13-s + 4·16-s + 19-s + 10i·28-s − 7·31-s − 10i·37-s − 5i·43-s − 18·49-s − 10i·52-s − 13·61-s + 8·64-s + 5i·67-s + 10i·73-s + ⋯ |
L(s) = 1 | + 4-s + 1.88i·7-s − 1.38i·13-s + 16-s + 0.229·19-s + 1.88i·28-s − 1.25·31-s − 1.64i·37-s − 0.762i·43-s − 2.57·49-s − 1.38i·52-s − 1.66·61-s + 64-s + 0.610i·67-s + 1.17i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44034 + 0.340019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44034 + 0.340019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 5iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.34436268877773826824390940586, −11.42765041955848287325130166705, −10.55139120736728572680468902968, −9.342256799032793945735448481535, −8.348809616381345941528354267912, −7.32422040193434457526132742217, −5.92646873641032548968684219242, −5.42886869452132221315686201870, −3.20639718690438586361714665153, −2.15579548274090720163004865639,
1.55320214804596082455230921611, 3.42938708516637227639041179406, 4.59633600622430725213246465689, 6.36362376933091395923522437303, 7.09361384705354003603112536606, 7.87110470045488970311903019173, 9.446957461320592505642302542068, 10.45407558964146273602629526461, 11.13023079679541338647056538270, 11.97213337598336306677428842685