| L(s) = 1 | + 1.58·3-s − 3.58·5-s − 7-s − 0.496·9-s − 11-s + 13-s − 5.67·15-s + 3.45·17-s − 2.87·19-s − 1.58·21-s − 3.76·23-s + 7.86·25-s − 5.53·27-s + 9.14·29-s − 10.0·31-s − 1.58·33-s + 3.58·35-s + 7.49·37-s + 1.58·39-s − 2.39·41-s − 9.24·43-s + 1.78·45-s + 3.77·47-s + 49-s + 5.46·51-s − 8.47·53-s + 3.58·55-s + ⋯ |
| L(s) = 1 | + 0.913·3-s − 1.60·5-s − 0.377·7-s − 0.165·9-s − 0.301·11-s + 0.277·13-s − 1.46·15-s + 0.838·17-s − 0.660·19-s − 0.345·21-s − 0.785·23-s + 1.57·25-s − 1.06·27-s + 1.69·29-s − 1.81·31-s − 0.275·33-s + 0.606·35-s + 1.23·37-s + 0.253·39-s − 0.373·41-s − 1.41·43-s + 0.265·45-s + 0.550·47-s + 0.142·49-s + 0.765·51-s − 1.16·53-s + 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.210204481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.210204481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 3.58T + 5T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 9.14T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 4.43T + 67T^{2} \) |
| 71 | \( 1 - 1.43T + 71T^{2} \) |
| 73 | \( 1 + 8.99T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 0.700T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 - 9.87T + 97T^{2} \) |
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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
−7.88001395137472344224611335014, −7.47181824570201480157264080634, −6.57555421298911781885911313936, −5.81894034899054561979112218083, −4.80824861943510999603454262021, −4.08221741068823030719301187794, −3.39181351670209485745540126183, −3.00699383718605237075866800692, −1.91362768146230456301565601918, −0.49925205202799189560543324035,
0.49925205202799189560543324035, 1.91362768146230456301565601918, 3.00699383718605237075866800692, 3.39181351670209485745540126183, 4.08221741068823030719301187794, 4.80824861943510999603454262021, 5.81894034899054561979112218083, 6.57555421298911781885911313936, 7.47181824570201480157264080634, 7.88001395137472344224611335014
