| L(s) = 1 | + 3-s + 0.416·5-s − 4.65·7-s + 9-s + 0.283·13-s + 0.416·15-s + 0.640·17-s + 5.88·19-s − 4.65·21-s − 5.88·23-s − 4.82·25-s + 27-s + 5.16·29-s − 1.22·31-s − 1.94·35-s − 6.82·37-s + 0.283·39-s + 12.4·41-s + 3.07·43-s + 0.416·45-s − 7.82·47-s + 14.7·49-s + 0.640·51-s − 5.46·53-s + 5.88·57-s − 3.20·59-s − 1.16·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.186·5-s − 1.76·7-s + 0.333·9-s + 0.0785·13-s + 0.107·15-s + 0.155·17-s + 1.35·19-s − 1.01·21-s − 1.22·23-s − 0.965·25-s + 0.192·27-s + 0.959·29-s − 0.220·31-s − 0.328·35-s − 1.12·37-s + 0.0453·39-s + 1.93·41-s + 0.469·43-s + 0.0620·45-s − 1.14·47-s + 2.10·49-s + 0.0897·51-s − 0.751·53-s + 0.779·57-s − 0.417·59-s − 0.149·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.836817742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836817742\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.416T + 5T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.283T + 13T^{2} \) |
| 17 | \( 1 - 0.640T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 + 5.88T + 23T^{2} \) |
| 29 | \( 1 - 5.16T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 53 | \( 1 + 5.46T + 53T^{2} \) |
| 59 | \( 1 + 3.20T + 59T^{2} \) |
| 61 | \( 1 + 1.16T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 - 8.60T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 + 9.27T + 83T^{2} \) |
| 89 | \( 1 - 3.94T + 89T^{2} \) |
| 97 | \( 1 - 8.99T + 97T^{2} \) |
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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
−7.71453985594708889637874880981, −7.25796442730082104047934227281, −6.29856190161680848847217512250, −6.03112279347668988134905783861, −5.06820664799839549207087459323, −4.01610641620631014532253822670, −3.42585427851412239403730044176, −2.83657092399691397297121727945, −1.91429545866810556926623857323, −0.63255404388318323404800543044,
0.63255404388318323404800543044, 1.91429545866810556926623857323, 2.83657092399691397297121727945, 3.42585427851412239403730044176, 4.01610641620631014532253822670, 5.06820664799839549207087459323, 6.03112279347668988134905783861, 6.29856190161680848847217512250, 7.25796442730082104047934227281, 7.71453985594708889637874880981
