| L(s) = 1 | − 9.89·5-s − 28·11-s + 4.24·13-s + 49.4·17-s − 5.65·19-s + 112·23-s − 27·25-s + 154·29-s − 33.9·31-s − 20·37-s + 168.·41-s − 76·43-s + 435.·47-s − 532·53-s + 277.·55-s + 316.·59-s − 168.·61-s − 41.9·65-s − 372·67-s − 168·71-s − 253.·73-s − 64·79-s − 673.·83-s − 490·85-s + 425.·89-s + 56.0·95-s + 1.05e3·97-s + ⋯ |
| L(s) = 1 | − 0.885·5-s − 0.767·11-s + 0.0905·13-s + 0.706·17-s − 0.0683·19-s + 1.01·23-s − 0.215·25-s + 0.986·29-s − 0.196·31-s − 0.0888·37-s + 0.641·41-s − 0.269·43-s + 1.35·47-s − 1.37·53-s + 0.679·55-s + 0.699·59-s − 0.353·61-s − 0.0801·65-s − 0.678·67-s − 0.280·71-s − 0.405·73-s − 0.0911·79-s − 0.890·83-s − 0.625·85-s + 0.506·89-s + 0.0604·95-s + 1.10·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 9.89T + 125T^{2} \) |
| 11 | \( 1 + 28T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.24T + 2.19e3T^{2} \) |
| 17 | \( 1 - 49.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.65T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154T + 2.43e4T^{2} \) |
| 31 | \( 1 + 33.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20T + 5.06e4T^{2} \) |
| 41 | \( 1 - 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 76T + 7.95e4T^{2} \) |
| 47 | \( 1 - 435.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 532T + 1.48e5T^{2} \) |
| 59 | \( 1 - 316.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 168.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 372T + 3.00e5T^{2} \) |
| 71 | \( 1 + 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + 253.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 64T + 4.93e5T^{2} \) |
| 83 | \( 1 + 673.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 425.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−8.429106190562277895876002125684, −7.73979074375626690756581231754, −7.16398840007416086785954526859, −6.11036224582449121211615394418, −5.20515703396227497864749363030, −4.38359034835442890463948635573, −3.42987321529970218521692859599, −2.59100770080960518020008213856, −1.13998528425414983558360732941, 0,
1.13998528425414983558360732941, 2.59100770080960518020008213856, 3.42987321529970218521692859599, 4.38359034835442890463948635573, 5.20515703396227497864749363030, 6.11036224582449121211615394418, 7.16398840007416086785954526859, 7.73979074375626690756581231754, 8.429106190562277895876002125684
