L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 2.41·11-s − 12-s − 13-s − 15-s + 16-s − 0.414·17-s − 18-s + 3.82·19-s + 20-s − 2.41·22-s − 8.65·23-s + 24-s − 4·25-s + 26-s − 27-s + 2.65·29-s + 30-s − 4.24·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.727·11-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 0.250·16-s − 0.100·17-s − 0.235·18-s + 0.878·19-s + 0.223·20-s − 0.514·22-s − 1.80·23-s + 0.204·24-s − 0.800·25-s + 0.196·26-s − 0.192·27-s + 0.493·29-s + 0.182·30-s − 0.762·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 17 | \( 1 + 0.414T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 8.65T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 5.07T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 0.585T + 89T^{2} \) |
| 97 | \( 1 - 7.31T + 97T^{2} \) |
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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
−7.979639116509307860586301761804, −7.54187931310204115467991397435, −6.54777418950223972095115932925, −6.07727344296996170865904915860, −5.32022954134583121949417701250, −4.31248723275319744200428680540, −3.39552933366423361009654806261, −2.15614595230913994198923739527, −1.36230987254613895198619212043, 0,
1.36230987254613895198619212043, 2.15614595230913994198923739527, 3.39552933366423361009654806261, 4.31248723275319744200428680540, 5.32022954134583121949417701250, 6.07727344296996170865904915860, 6.54777418950223972095115932925, 7.54187931310204115467991397435, 7.979639116509307860586301761804