| L(s) = 1 | − 3-s − 0.554·5-s − 3.04·7-s + 9-s + 1.80·11-s + 0.554·15-s + 1.24·17-s + 2.35·19-s + 3.04·21-s − 0.554·23-s − 4.69·25-s − 27-s − 6.18·29-s + 8.67·31-s − 1.80·33-s + 1.69·35-s + 0.960·37-s − 2.47·41-s + 0.384·43-s − 0.554·45-s + 9.96·47-s + 2.29·49-s − 1.24·51-s + 6.02·53-s − 55-s − 2.35·57-s − 7.30·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.248·5-s − 1.15·7-s + 0.333·9-s + 0.543·11-s + 0.143·15-s + 0.302·17-s + 0.540·19-s + 0.665·21-s − 0.115·23-s − 0.938·25-s − 0.192·27-s − 1.14·29-s + 1.55·31-s − 0.313·33-s + 0.286·35-s + 0.157·37-s − 0.386·41-s + 0.0585·43-s − 0.0827·45-s + 1.45·47-s + 0.327·49-s − 0.174·51-s + 0.827·53-s − 0.134·55-s − 0.312·57-s − 0.951·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.554T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 0.554T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 - 0.960T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 0.384T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 - 6.02T + 53T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 4.58T + 71T^{2} \) |
| 73 | \( 1 - 1.04T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.920471703191222657266274390824, −7.28717237583194305453040765450, −6.47708417210960459576898683745, −5.98692533917237083518478025116, −5.18391553936642648646394872229, −4.13424296487479626499647633025, −3.55140853951662204956228032409, −2.55026159575189952359999702535, −1.20664881988740663977941205003, 0,
1.20664881988740663977941205003, 2.55026159575189952359999702535, 3.55140853951662204956228032409, 4.13424296487479626499647633025, 5.18391553936642648646394872229, 5.98692533917237083518478025116, 6.47708417210960459576898683745, 7.28717237583194305453040765450, 7.920471703191222657266274390824
