| L(s) = 1 | − 1.41·5-s + 4·11-s + 4.24·13-s − 1.41·17-s − 2.82·19-s − 4·23-s − 2.99·25-s − 8·29-s − 8·37-s + 7.07·41-s + 4·43-s + 5.65·47-s − 10·53-s − 5.65·55-s + 14.1·59-s − 7.07·61-s − 6·65-s − 7.07·73-s − 8·79-s − 14.1·83-s + 2.00·85-s − 7.07·89-s + 4.00·95-s − 1.41·97-s + 12.7·101-s + 11.3·103-s + 8·107-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + 1.20·11-s + 1.17·13-s − 0.342·17-s − 0.648·19-s − 0.834·23-s − 0.599·25-s − 1.48·29-s − 1.31·37-s + 1.10·41-s + 0.609·43-s + 0.825·47-s − 1.37·53-s − 0.762·55-s + 1.84·59-s − 0.905·61-s − 0.744·65-s − 0.827·73-s − 0.900·79-s − 1.55·83-s + 0.216·85-s − 0.749·89-s + 0.410·95-s − 0.143·97-s + 1.26·101-s + 1.11·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 1.41T + 97T^{2} \) |
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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
−7.54647306064248146936788261911, −6.96148588433015862403679778596, −6.08936732763621987205476119746, −5.73730804514242948151315572227, −4.44210680492621073787956375774, −3.95406194014190649629836945849, −3.44215705137856847338883170986, −2.14967554283221754490742981265, −1.31318412163359650684991690255, 0,
1.31318412163359650684991690255, 2.14967554283221754490742981265, 3.44215705137856847338883170986, 3.95406194014190649629836945849, 4.44210680492621073787956375774, 5.73730804514242948151315572227, 6.08936732763621987205476119746, 6.96148588433015862403679778596, 7.54647306064248146936788261911
