| L(s) = 1 | + 0.792·2-s − 1.37·4-s + 3.46·7-s − 2.67·8-s − 4.37·11-s + 5.84·13-s + 2.74·14-s + 0.627·16-s + 0.792·17-s − 0.372·19-s − 3.46·22-s + 1.58·23-s + 4.62·26-s − 4.75·28-s − 5.74·29-s + 6.37·31-s + 5.84·32-s + 0.627·34-s − 2.37·37-s − 0.294·38-s + 4.37·41-s − 3.46·43-s + 6·44-s + 1.25·46-s + 1.87·47-s + 4.99·49-s − 8.01·52-s + ⋯ |
| L(s) = 1 | + 0.560·2-s − 0.686·4-s + 1.30·7-s − 0.944·8-s − 1.31·11-s + 1.61·13-s + 0.733·14-s + 0.156·16-s + 0.192·17-s − 0.0854·19-s − 0.738·22-s + 0.330·23-s + 0.907·26-s − 0.898·28-s − 1.06·29-s + 1.14·31-s + 1.03·32-s + 0.107·34-s − 0.390·37-s − 0.0478·38-s + 0.682·41-s − 0.528·43-s + 0.904·44-s + 0.185·46-s + 0.274·47-s + 0.714·49-s − 1.11·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.142102498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.142102498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 - 0.792T + 17T^{2} \) |
| 19 | \( 1 + 0.372T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 5.74T + 29T^{2} \) |
| 31 | \( 1 - 6.37T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 2.37T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−8.949150875650240791462667651737, −8.260408546580606006626063804682, −7.911844990781217766881305029940, −6.65768787783319802393800337747, −5.53719477193458471242261754997, −5.25342716471809768409104972232, −4.27391875959344170045112216843, −3.52526419063155925093872385580, −2.32371804454322799126956888294, −0.938050921581072324313989381296,
0.938050921581072324313989381296, 2.32371804454322799126956888294, 3.52526419063155925093872385580, 4.27391875959344170045112216843, 5.25342716471809768409104972232, 5.53719477193458471242261754997, 6.65768787783319802393800337747, 7.911844990781217766881305029940, 8.260408546580606006626063804682, 8.949150875650240791462667651737
