L(s) = 1 | + 1.68·2-s − 3-s + 0.841·4-s − 1.68·6-s − 1.95·8-s + 9-s + 1.77·11-s − 0.841·12-s + 3.05·13-s − 4.97·16-s − 7.59·17-s + 1.68·18-s − 3.12·19-s + 2.99·22-s + 4.18·23-s + 1.95·24-s + 5.14·26-s − 27-s + 7.08·29-s − 0.871·31-s − 4.47·32-s − 1.77·33-s − 12.8·34-s + 0.841·36-s − 9.91·37-s − 5.27·38-s − 3.05·39-s + ⋯ |
L(s) = 1 | + 1.19·2-s − 0.577·3-s + 0.420·4-s − 0.688·6-s − 0.690·8-s + 0.333·9-s + 0.535·11-s − 0.242·12-s + 0.846·13-s − 1.24·16-s − 1.84·17-s + 0.397·18-s − 0.717·19-s + 0.637·22-s + 0.871·23-s + 0.398·24-s + 1.00·26-s − 0.192·27-s + 1.31·29-s − 0.156·31-s − 0.791·32-s − 0.309·33-s − 2.19·34-s + 0.140·36-s − 1.62·37-s − 0.855·38-s − 0.488·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.68T + 2T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 4.18T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 + 0.871T + 31T^{2} \) |
| 37 | \( 1 + 9.91T + 37T^{2} \) |
| 41 | \( 1 - 2.76T + 41T^{2} \) |
| 43 | \( 1 + 0.317T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 6.13T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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
−8.274894960989520085083315643579, −6.82521912299958753691976466513, −6.59085296736984534277116262777, −5.90575669704758819388926021958, −4.91362837602413955301425931613, −4.49104158069676544093242873571, −3.70254619828917951909931896152, −2.78719161152510199587563335430, −1.58268706753623652450412379645, 0,
1.58268706753623652450412379645, 2.78719161152510199587563335430, 3.70254619828917951909931896152, 4.49104158069676544093242873571, 4.91362837602413955301425931613, 5.90575669704758819388926021958, 6.59085296736984534277116262777, 6.82521912299958753691976466513, 8.274894960989520085083315643579