| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 11-s − 1.58·13-s + 15-s − 3.16·17-s + 1.10·19-s + 21-s − 5.42·23-s + 25-s + 27-s + 6.74·29-s − 1.58·31-s + 33-s + 35-s + 0.682·37-s − 1.58·39-s + 1.31·41-s + 12.8·43-s + 45-s + 12.1·47-s + 49-s − 3.16·51-s + 10.3·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s − 0.439·13-s + 0.258·15-s − 0.767·17-s + 0.253·19-s + 0.218·21-s − 1.13·23-s + 0.200·25-s + 0.192·27-s + 1.25·29-s − 0.284·31-s + 0.174·33-s + 0.169·35-s + 0.112·37-s − 0.253·39-s + 0.205·41-s + 1.95·43-s + 0.149·45-s + 1.77·47-s + 0.142·49-s − 0.442·51-s + 1.41·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.820840555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.820840555\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 - 1.10T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 6.74T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 - 0.682T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 0.420T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 + 2.05T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 7.64T + 97T^{2} \) |
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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
−8.446588733644614310143470634206, −7.57029195430630442727764833299, −7.00265759954021870253582640312, −6.12140392202819132307430069391, −5.41614656350798592698496067720, −4.42744083261576078887399222044, −3.90227078101038035249225649930, −2.63977788586966762064941485662, −2.14583495643937575801979299379, −0.928590532592624967120233297396,
0.928590532592624967120233297396, 2.14583495643937575801979299379, 2.63977788586966762064941485662, 3.90227078101038035249225649930, 4.42744083261576078887399222044, 5.41614656350798592698496067720, 6.12140392202819132307430069391, 7.00265759954021870253582640312, 7.57029195430630442727764833299, 8.446588733644614310143470634206
