| L(s) = 1 | − 3-s + 5-s + 4.60·7-s + 9-s − 4.60·11-s − 15-s − 5.60·17-s + 0.605·19-s − 4.60·21-s + 4.60·23-s − 4·25-s − 27-s + 3·29-s − 9.21·31-s + 4.60·33-s + 4.60·35-s + 9.60·37-s + 4.39·41-s − 4.60·43-s + 45-s − 8.60·47-s + 14.2·49-s + 5.60·51-s + 3·53-s − 4.60·55-s − 0.605·57-s − 5.21·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.74·7-s + 0.333·9-s − 1.38·11-s − 0.258·15-s − 1.35·17-s + 0.138·19-s − 1.00·21-s + 0.960·23-s − 0.800·25-s − 0.192·27-s + 0.557·29-s − 1.65·31-s + 0.801·33-s + 0.778·35-s + 1.57·37-s + 0.686·41-s − 0.702·43-s + 0.149·45-s − 1.25·47-s + 2.03·49-s + 0.784·51-s + 0.412·53-s − 0.621·55-s − 0.0802·57-s − 0.678·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 4.60T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 9.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 + 4.60T + 43T^{2} \) |
| 47 | \( 1 + 8.60T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 5.21T + 59T^{2} \) |
| 61 | \( 1 + 0.394T + 61T^{2} \) |
| 67 | \( 1 + 3.39T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 9.81T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 3.21T + 97T^{2} \) |
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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
−7.55326992612644284338513602082, −6.85488902078981957293180013481, −5.84800089670536786755329886231, −5.41383843189906799581102611569, −4.69159011119179330232435529097, −4.30340060248325367982107153647, −2.88937917399538163806151746224, −2.08876253186317585583360738016, −1.37162036043277068462618055738, 0,
1.37162036043277068462618055738, 2.08876253186317585583360738016, 2.88937917399538163806151746224, 4.30340060248325367982107153647, 4.69159011119179330232435529097, 5.41383843189906799581102611569, 5.84800089670536786755329886231, 6.85488902078981957293180013481, 7.55326992612644284338513602082
