L(s) = 1 | − 1.83·2-s + 3-s + 1.35·4-s + 0.943·5-s − 1.83·6-s + 1.17·8-s + 9-s − 1.72·10-s + 2.08·11-s + 1.35·12-s − 3.64·13-s + 0.943·15-s − 4.87·16-s − 0.299·17-s − 1.83·18-s + 0.579·19-s + 1.28·20-s − 3.82·22-s − 23-s + 1.17·24-s − 4.10·25-s + 6.67·26-s + 27-s − 5.89·29-s − 1.72·30-s + 6.02·31-s + 6.57·32-s + ⋯ |
L(s) = 1 | − 1.29·2-s + 0.577·3-s + 0.679·4-s + 0.421·5-s − 0.748·6-s + 0.415·8-s + 0.333·9-s − 0.546·10-s + 0.629·11-s + 0.392·12-s − 1.01·13-s + 0.243·15-s − 1.21·16-s − 0.0726·17-s − 0.431·18-s + 0.132·19-s + 0.286·20-s − 0.815·22-s − 0.208·23-s + 0.239·24-s − 0.821·25-s + 1.30·26-s + 0.192·27-s − 1.09·29-s − 0.315·30-s + 1.08·31-s + 1.16·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 0.943T + 5T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 + 0.299T + 17T^{2} \) |
| 19 | \( 1 - 0.579T + 19T^{2} \) |
| 29 | \( 1 + 5.89T + 29T^{2} \) |
| 31 | \( 1 - 6.02T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 + 5.06T + 47T^{2} \) |
| 53 | \( 1 - 2.32T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 4.13T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 2.67T + 89T^{2} \) |
| 97 | \( 1 + 0.589T + 97T^{2} \) |
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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
−8.321495573400611518096018210286, −7.73090806524118656132341134208, −7.02716590130196828071415025152, −6.31622749478835677921168384799, −5.15139569057550093795348392691, −4.34585138792910028959781665189, −3.28055853256049087130739499772, −2.14172167037867203221483257159, −1.48557072580709489576883139560, 0,
1.48557072580709489576883139560, 2.14172167037867203221483257159, 3.28055853256049087130739499772, 4.34585138792910028959781665189, 5.15139569057550093795348392691, 6.31622749478835677921168384799, 7.02716590130196828071415025152, 7.73090806524118656132341134208, 8.321495573400611518096018210286