| L(s) = 1 | + 2-s + 4-s − 3.37·7-s + 8-s − 4.37·11-s + 6.74·13-s − 3.37·14-s + 16-s + 1.62·17-s − 2.37·19-s − 4.37·22-s − 1.37·23-s + 6.74·26-s − 3.37·28-s + 1.37·29-s + 4.74·31-s + 32-s + 1.62·34-s + 4·37-s − 2.37·38-s − 3·41-s + 5.62·43-s − 4.37·44-s − 1.37·46-s + 7.37·47-s + 4.37·49-s + 6.74·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.27·7-s + 0.353·8-s − 1.31·11-s + 1.87·13-s − 0.901·14-s + 0.250·16-s + 0.394·17-s − 0.544·19-s − 0.932·22-s − 0.286·23-s + 1.32·26-s − 0.637·28-s + 0.254·29-s + 0.852·31-s + 0.176·32-s + 0.279·34-s + 0.657·37-s − 0.384·38-s − 0.468·41-s + 0.858·43-s − 0.659·44-s − 0.202·46-s + 1.07·47-s + 0.624·49-s + 0.935·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.523792573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.523792573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 - 1.62T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 5.62T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 - 7T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 3.11T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
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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
−8.393654551240342517120479518039, −7.63630938807426205208881495949, −6.77219343662329818992683449477, −6.01520250820010483849625892974, −5.73264286940115781230820259714, −4.58719207551464155698526797573, −3.75198076459123684632222357284, −3.10430723010243721441050799738, −2.28268008657432106710816115058, −0.805764038373834662278906042714,
0.805764038373834662278906042714, 2.28268008657432106710816115058, 3.10430723010243721441050799738, 3.75198076459123684632222357284, 4.58719207551464155698526797573, 5.73264286940115781230820259714, 6.01520250820010483849625892974, 6.77219343662329818992683449477, 7.63630938807426205208881495949, 8.393654551240342517120479518039
