| L(s) = 1 | − 3-s + 5-s − 3.30·7-s + 9-s + 2.24·11-s − 3.19·13-s − 15-s + 3.55·17-s − 19-s + 3.30·21-s + 1.55·23-s + 25-s − 27-s + 4.24·29-s − 7.43·31-s − 2.24·33-s − 3.30·35-s − 5.30·37-s + 3.19·39-s + 2.36·41-s + 3.80·43-s + 45-s − 9.55·47-s + 3.94·49-s − 3.55·51-s − 3.55·53-s + 2.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.25·7-s + 0.333·9-s + 0.678·11-s − 0.884·13-s − 0.258·15-s + 0.862·17-s − 0.229·19-s + 0.721·21-s + 0.324·23-s + 0.200·25-s − 0.192·27-s + 0.789·29-s − 1.33·31-s − 0.391·33-s − 0.559·35-s − 0.872·37-s + 0.510·39-s + 0.369·41-s + 0.580·43-s + 0.149·45-s − 1.39·47-s + 0.563·49-s − 0.498·51-s − 0.488·53-s + 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 7.43T + 31T^{2} \) |
| 37 | \( 1 + 5.30T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 + 9.55T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 - 0.498T + 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 0.117T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 7.68T + 97T^{2} \) |
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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
−8.899944735596474152475211232323, −7.66073330845500757154470076045, −6.92140919991834372947016840676, −6.30126094896345817201101982184, −5.58808063198434726787102923828, −4.72455795608325177576362039841, −3.64574304128244307403457979507, −2.79622383721541992509548105205, −1.46378896931079200813069068779, 0,
1.46378896931079200813069068779, 2.79622383721541992509548105205, 3.64574304128244307403457979507, 4.72455795608325177576362039841, 5.58808063198434726787102923828, 6.30126094896345817201101982184, 6.92140919991834372947016840676, 7.66073330845500757154470076045, 8.899944735596474152475211232323
