| L(s) = 1 | − 3.46·7-s − 6.92·13-s + 8·19-s − 5·25-s + 10.3·31-s + 6.92·37-s + 8·43-s + 4.99·49-s + 6.92·61-s + 16·67-s + 10·73-s − 17.3·79-s + 23.9·91-s − 14·97-s + 3.46·103-s + 20.7·109-s + ⋯ |
| L(s) = 1 | − 1.30·7-s − 1.92·13-s + 1.83·19-s − 25-s + 1.86·31-s + 1.13·37-s + 1.21·43-s + 0.714·49-s + 0.887·61-s + 1.95·67-s + 1.17·73-s − 1.94·79-s + 2.51·91-s − 1.42·97-s + 0.341·103-s + 1.99·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.235558073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.235558073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.92T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 - 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.332707224469189135706370334204, −8.090352873655926573514295386900, −7.41600527888526306211129247287, −6.76620814624473543962950074902, −5.87637448564186668177525096222, −5.09438222617675459139753777471, −4.13212158993635176805229890657, −3.05803730692776013728153443185, −2.44090522270230126749124540829, −0.69332551679183177694184092560,
0.69332551679183177694184092560, 2.44090522270230126749124540829, 3.05803730692776013728153443185, 4.13212158993635176805229890657, 5.09438222617675459139753777471, 5.87637448564186668177525096222, 6.76620814624473543962950074902, 7.41600527888526306211129247287, 8.090352873655926573514295386900, 9.332707224469189135706370334204
