| L(s) = 1 | + 2.07·2-s − 3-s + 2.31·4-s − 1.54·5-s − 2.07·6-s − 2.97·7-s + 0.657·8-s + 9-s − 3.21·10-s − 2.31·12-s − 0.922·13-s − 6.17·14-s + 1.54·15-s − 3.26·16-s + 4.29·17-s + 2.07·18-s − 19-s − 3.58·20-s + 2.97·21-s + 1.21·23-s − 0.657·24-s − 2.60·25-s − 1.91·26-s − 27-s − 6.89·28-s − 3.55·29-s + 3.21·30-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 0.577·3-s + 1.15·4-s − 0.691·5-s − 0.848·6-s − 1.12·7-s + 0.232·8-s + 0.333·9-s − 1.01·10-s − 0.668·12-s − 0.255·13-s − 1.65·14-s + 0.399·15-s − 0.816·16-s + 1.04·17-s + 0.489·18-s − 0.229·19-s − 0.801·20-s + 0.649·21-s + 0.253·23-s − 0.134·24-s − 0.521·25-s − 0.375·26-s − 0.192·27-s − 1.30·28-s − 0.660·29-s + 0.586·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015527997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015527997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.07T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 + 2.97T + 7T^{2} \) |
| 13 | \( 1 + 0.922T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 3.55T + 29T^{2} \) |
| 31 | \( 1 - 1.76T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 + 0.365T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61952012275402283310534949986, −7.01742866529617780526351492387, −6.30090645331188445429092451618, −5.80665884364432258458797602313, −5.09233010222046160237817135698, −4.36799107877078587900864224257, −3.60909225474907473493957955543, −3.21145377367822069388822173448, −2.15521774505842376492931706335, −0.56922600417746359673095096126,
0.56922600417746359673095096126, 2.15521774505842376492931706335, 3.21145377367822069388822173448, 3.60909225474907473493957955543, 4.36799107877078587900864224257, 5.09233010222046160237817135698, 5.80665884364432258458797602313, 6.30090645331188445429092451618, 7.01742866529617780526351492387, 7.61952012275402283310534949986
