| L(s) = 1 | + 3-s − 2.71·5-s − 0.775·7-s + 9-s + 3.70·11-s − 6.16·13-s − 2.71·15-s + 6.36·17-s + 0.922·19-s − 0.775·21-s − 4.47·23-s + 2.34·25-s + 27-s − 5.18·29-s + 1.35·31-s + 3.70·33-s + 2.10·35-s + 11.5·37-s − 6.16·39-s − 8.90·41-s + 5.52·43-s − 2.71·45-s + 8.07·47-s − 6.39·49-s + 6.36·51-s − 2.23·53-s − 10.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.21·5-s − 0.293·7-s + 0.333·9-s + 1.11·11-s − 1.71·13-s − 0.699·15-s + 1.54·17-s + 0.211·19-s − 0.169·21-s − 0.932·23-s + 0.469·25-s + 0.192·27-s − 0.962·29-s + 0.243·31-s + 0.645·33-s + 0.355·35-s + 1.89·37-s − 0.987·39-s − 1.39·41-s + 0.841·43-s − 0.404·45-s + 1.17·47-s − 0.914·49-s + 0.891·51-s − 0.307·53-s − 1.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637669761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637669761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 + 0.775T + 7T^{2} \) |
| 11 | \( 1 - 3.70T + 11T^{2} \) |
| 13 | \( 1 + 6.16T + 13T^{2} \) |
| 17 | \( 1 - 6.36T + 17T^{2} \) |
| 19 | \( 1 - 0.922T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 5.18T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 11.5T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 - 5.52T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 + 2.23T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 5.08T + 61T^{2} \) |
| 67 | \( 1 - 0.628T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 3.83T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 4.76T + 83T^{2} \) |
| 89 | \( 1 - 0.390T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.226234431196161308697507523886, −7.69313306296803867388709203691, −7.29691968146935670694207431090, −6.38094660987044428305232148230, −5.39430354809866199593036974699, −4.42903929536760003603788383358, −3.79867819641083062630371713778, −3.13608283206431732014312605964, −2.07690707796443274471626504519, −0.70519871640820198692633986713,
0.70519871640820198692633986713, 2.07690707796443274471626504519, 3.13608283206431732014312605964, 3.79867819641083062630371713778, 4.42903929536760003603788383358, 5.39430354809866199593036974699, 6.38094660987044428305232148230, 7.29691968146935670694207431090, 7.69313306296803867388709203691, 8.226234431196161308697507523886
