L(s) = 1 | + 2.23·2-s + 1.36·3-s + 2.98·4-s + 4.28·5-s + 3.03·6-s + 0.384·7-s + 2.19·8-s − 1.14·9-s + 9.56·10-s + 11-s + 4.06·12-s + 0.857·14-s + 5.83·15-s − 1.07·16-s − 4.75·17-s − 2.55·18-s − 2.17·19-s + 12.7·20-s + 0.523·21-s + 2.23·22-s + 2.70·23-s + 2.98·24-s + 13.3·25-s − 5.64·27-s + 1.14·28-s − 10.2·29-s + 13.0·30-s + ⋯ |
L(s) = 1 | + 1.57·2-s + 0.786·3-s + 1.49·4-s + 1.91·5-s + 1.24·6-s + 0.145·7-s + 0.774·8-s − 0.381·9-s + 3.02·10-s + 0.301·11-s + 1.17·12-s + 0.229·14-s + 1.50·15-s − 0.268·16-s − 1.15·17-s − 0.602·18-s − 0.498·19-s + 2.85·20-s + 0.114·21-s + 0.475·22-s + 0.563·23-s + 0.609·24-s + 2.67·25-s − 1.08·27-s + 0.216·28-s − 1.90·29-s + 2.37·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.928316829\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.928316829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 - 4.28T + 5T^{2} \) |
| 7 | \( 1 - 0.384T + 7T^{2} \) |
| 17 | \( 1 + 4.75T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 2.70T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 + 4.30T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 0.639T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 3.32T + 53T^{2} \) |
| 59 | \( 1 + 8.90T + 59T^{2} \) |
| 61 | \( 1 - 8.06T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 7.58T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−9.014507219833987772454102698987, −8.848167797346294120093108735946, −7.34255557519508144525001258901, −6.45663555688282856725591836301, −5.90383405778042051701899428374, −5.23092620178189887859909950443, −4.34588800322975813728638896069, −3.27390286921320943536555635799, −2.40581902490591092453779849779, −1.88238114021611440476916185999,
1.88238114021611440476916185999, 2.40581902490591092453779849779, 3.27390286921320943536555635799, 4.34588800322975813728638896069, 5.23092620178189887859909950443, 5.90383405778042051701899428374, 6.45663555688282856725591836301, 7.34255557519508144525001258901, 8.848167797346294120093108735946, 9.014507219833987772454102698987