| L(s) = 1 | − 3-s − 5-s + 1.41·7-s + 9-s + 2.24·11-s − 3.41·13-s + 15-s + 1.17·17-s + 19-s − 1.41·21-s − 7.65·23-s + 25-s − 27-s + 1.41·29-s + 3.17·31-s − 2.24·33-s − 1.41·35-s − 3.41·37-s + 3.41·39-s − 0.242·41-s − 12.2·43-s − 45-s + 7.65·47-s − 5·49-s − 1.17·51-s + 8·53-s − 2.24·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.534·7-s + 0.333·9-s + 0.676·11-s − 0.946·13-s + 0.258·15-s + 0.284·17-s + 0.229·19-s − 0.308·21-s − 1.59·23-s + 0.200·25-s − 0.192·27-s + 0.262·29-s + 0.569·31-s − 0.390·33-s − 0.239·35-s − 0.561·37-s + 0.546·39-s − 0.0378·41-s − 1.86·43-s − 0.149·45-s + 1.11·47-s − 0.714·49-s − 0.164·51-s + 1.09·53-s − 0.302·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41T + 37T^{2} \) |
| 41 | \( 1 + 0.242T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 5.89T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−8.000044225901767224227961642725, −7.17848085074397315353680118348, −6.59049727721469305012632514707, −5.70646453267605422761103764100, −4.97975539405571724413945817985, −4.28789627647524852567730305147, −3.51832248531185203347291894514, −2.31678414121467265869669339776, −1.30286531081669497257238006184, 0,
1.30286531081669497257238006184, 2.31678414121467265869669339776, 3.51832248531185203347291894514, 4.28789627647524852567730305147, 4.97975539405571724413945817985, 5.70646453267605422761103764100, 6.59049727721469305012632514707, 7.17848085074397315353680118348, 8.000044225901767224227961642725
