| L(s) = 1 | − 2·3-s − 5-s + 9-s − 5.27·11-s + 2·13-s + 2·15-s − 0.274·17-s − 19-s + 7.54·23-s − 4·25-s + 4·27-s − 4·29-s − 4·31-s + 10.5·33-s + 2·37-s − 4·39-s + 10.5·41-s − 43-s − 45-s + 5.27·47-s + 0.549·51-s + 8.54·53-s + 5.27·55-s + 2·57-s − 4·59-s + 4.72·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.333·9-s − 1.59·11-s + 0.554·13-s + 0.516·15-s − 0.0666·17-s − 0.229·19-s + 1.57·23-s − 0.800·25-s + 0.769·27-s − 0.742·29-s − 0.718·31-s + 1.83·33-s + 0.328·37-s − 0.640·39-s + 1.64·41-s − 0.152·43-s − 0.149·45-s + 0.769·47-s + 0.0769·51-s + 1.17·53-s + 0.711·55-s + 0.264·57-s − 0.520·59-s + 0.604·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 0.274T + 17T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4.54T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 + 6.54T + 97T^{2} \) |
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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
−7.41844493327151496567885410687, −6.92928575005735506097120060323, −5.79792728434284334003909875474, −5.66423659455369253905440290275, −4.84320561306404727131219965527, −4.10863472142883553969771089425, −3.13012576482682043869654226361, −2.28638946275807995511729319729, −0.925657262824924561006678775950, 0,
0.925657262824924561006678775950, 2.28638946275807995511729319729, 3.13012576482682043869654226361, 4.10863472142883553969771089425, 4.84320561306404727131219965527, 5.66423659455369253905440290275, 5.79792728434284334003909875474, 6.92928575005735506097120060323, 7.41844493327151496567885410687
