| L(s) = 1 | + 2-s + 4-s + 2.82·5-s + 8-s + 2.82·10-s − 11-s + 4.24·13-s + 16-s + 5.65·17-s + 7.07·19-s + 2.82·20-s − 22-s + 3.00·25-s + 4.24·26-s − 8·29-s + 1.41·31-s + 32-s + 5.65·34-s + 6·37-s + 7.07·38-s + 2.82·40-s + 4·43-s − 44-s − 7.07·47-s + 3.00·50-s + 4.24·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.26·5-s + 0.353·8-s + 0.894·10-s − 0.301·11-s + 1.17·13-s + 0.250·16-s + 1.37·17-s + 1.62·19-s + 0.632·20-s − 0.213·22-s + 0.600·25-s + 0.832·26-s − 1.48·29-s + 0.254·31-s + 0.176·32-s + 0.970·34-s + 0.986·37-s + 1.14·38-s + 0.447·40-s + 0.609·43-s − 0.150·44-s − 1.03·47-s + 0.424·50-s + 0.588·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.325291978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.325291978\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 7.07T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 7.07T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
| 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
−7.63362238073388942311955634834, −6.85653956103152510656696446210, −5.98614230469501452683895281319, −5.66037853163319986170706656983, −5.21461651676620261229250353823, −4.15724220995996450773411233157, −3.35297812324539984375493290858, −2.77528852017348765490566550807, −1.71261883885942142666576884402, −1.09588561555603425131919303949,
1.09588561555603425131919303949, 1.71261883885942142666576884402, 2.77528852017348765490566550807, 3.35297812324539984375493290858, 4.15724220995996450773411233157, 5.21461651676620261229250353823, 5.66037853163319986170706656983, 5.98614230469501452683895281319, 6.85653956103152510656696446210, 7.63362238073388942311955634834
