| L(s) = 1 | + 1.69·3-s − 1.37·5-s + 4.77·7-s − 0.111·9-s − 13-s − 2.33·15-s + 1.82·17-s − 3.82·19-s + 8.11·21-s − 0.328·23-s − 3.10·25-s − 5.28·27-s − 1.40·29-s + 6.79·31-s − 6.56·35-s − 8.01·37-s − 1.69·39-s + 11.8·41-s + 3.92·43-s + 0.153·45-s + 6.43·47-s + 15.7·49-s + 3.10·51-s + 13.0·53-s − 6.50·57-s + 8.42·59-s + 9.90·61-s + ⋯ |
| L(s) = 1 | + 0.981·3-s − 0.615·5-s + 1.80·7-s − 0.0370·9-s − 0.277·13-s − 0.603·15-s + 0.442·17-s − 0.877·19-s + 1.76·21-s − 0.0685·23-s − 0.621·25-s − 1.01·27-s − 0.260·29-s + 1.21·31-s − 1.10·35-s − 1.31·37-s − 0.272·39-s + 1.85·41-s + 0.598·43-s + 0.0228·45-s + 0.938·47-s + 2.25·49-s + 0.434·51-s + 1.79·53-s − 0.861·57-s + 1.09·59-s + 1.26·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.060125236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.060125236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 + 0.328T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 - 6.79T + 31T^{2} \) |
| 37 | \( 1 + 8.01T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 9.90T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 - 0.651T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 + 5.75T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.132061009053912873558008749614, −7.62640476566572145657653522521, −6.92459582127180230049337216765, −5.71433806359132502342263967011, −5.15343169549535593361149743749, −4.12054880634934086450908493100, −3.87276380210863305778264791323, −2.52542227463087324950141794581, −2.11330584796947063728489795488, −0.881825890426412755718319151976,
0.881825890426412755718319151976, 2.11330584796947063728489795488, 2.52542227463087324950141794581, 3.87276380210863305778264791323, 4.12054880634934086450908493100, 5.15343169549535593361149743749, 5.71433806359132502342263967011, 6.92459582127180230049337216765, 7.62640476566572145657653522521, 8.132061009053912873558008749614
