| L(s) = 1 | − 3.26·3-s − 1.04·5-s − 7-s + 7.68·9-s − 3.26·11-s + 13-s + 3.41·15-s − 7.95·17-s − 3.04·19-s + 3.26·21-s + 6.31·23-s − 3.90·25-s − 15.3·27-s − 4.46·29-s − 3.19·31-s + 10.6·33-s + 1.04·35-s − 1.26·37-s − 3.26·39-s + 10.7·41-s − 6.90·43-s − 8.02·45-s − 3.19·47-s + 49-s + 25.9·51-s + 8.46·53-s + 3.41·55-s + ⋯ |
| L(s) = 1 | − 1.88·3-s − 0.467·5-s − 0.377·7-s + 2.56·9-s − 0.985·11-s + 0.277·13-s + 0.881·15-s − 1.92·17-s − 0.698·19-s + 0.713·21-s + 1.31·23-s − 0.781·25-s − 2.94·27-s − 0.828·29-s − 0.573·31-s + 1.85·33-s + 0.176·35-s − 0.208·37-s − 0.523·39-s + 1.68·41-s − 1.05·43-s − 1.19·45-s − 0.465·47-s + 0.142·49-s + 3.64·51-s + 1.16·53-s + 0.460·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3724311579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3724311579\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 17 | \( 1 + 7.95T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 1.26T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 + 3.19T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 0.146T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 - 2.66T + 89T^{2} \) |
| 97 | \( 1 + 1.34T + 97T^{2} \) |
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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
−9.719247341532428432628092457467, −8.796175520149143960943277151970, −7.61554530085468588505837181711, −6.88213755786622957535869439596, −6.24462519678326743837129932876, −5.38146763886276919053609789184, −4.65769271370235028836318429980, −3.81697433782169739237850091304, −2.13419392558698429702889210347, −0.45879166586907633586289008137,
0.45879166586907633586289008137, 2.13419392558698429702889210347, 3.81697433782169739237850091304, 4.65769271370235028836318429980, 5.38146763886276919053609789184, 6.24462519678326743837129932876, 6.88213755786622957535869439596, 7.61554530085468588505837181711, 8.796175520149143960943277151970, 9.719247341532428432628092457467
