L(s) = 1 | + 2-s − 2.25·3-s + 4-s + 1.19·5-s − 2.25·6-s + 7-s + 8-s + 2.07·9-s + 1.19·10-s + 3.37·11-s − 2.25·12-s + 3.77·13-s + 14-s − 2.70·15-s + 16-s + 8.18·17-s + 2.07·18-s + 1.19·20-s − 2.25·21-s + 3.37·22-s + 7.97·23-s − 2.25·24-s − 3.56·25-s + 3.77·26-s + 2.07·27-s + 28-s + 2.44·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.30·3-s + 0.5·4-s + 0.536·5-s − 0.919·6-s + 0.377·7-s + 0.353·8-s + 0.692·9-s + 0.379·10-s + 1.01·11-s − 0.650·12-s + 1.04·13-s + 0.267·14-s − 0.697·15-s + 0.250·16-s + 1.98·17-s + 0.489·18-s + 0.268·20-s − 0.491·21-s + 0.718·22-s + 1.66·23-s − 0.459·24-s − 0.712·25-s + 0.740·26-s + 0.399·27-s + 0.188·28-s + 0.454·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.925774206\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.925774206\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 8.18T + 17T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 4.06T + 37T^{2} \) |
| 41 | \( 1 + 9.10T + 41T^{2} \) |
| 43 | \( 1 + 6.90T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 - 8.79T + 53T^{2} \) |
| 59 | \( 1 + 7.26T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 - 8.97T + 71T^{2} \) |
| 73 | \( 1 + 8.21T + 73T^{2} \) |
| 79 | \( 1 - 0.0994T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.162887833307357061119312635822, −7.11072838228820363744423359939, −6.62092155118167853349415996974, −5.81835871848670283166700104716, −5.47215373239815982606467424212, −4.81121049574883958779355096319, −3.79249348248050154554852468057, −3.12143228303703920949314958150, −1.60667010917084266975740644222, −1.01915981413939143967411072944,
1.01915981413939143967411072944, 1.60667010917084266975740644222, 3.12143228303703920949314958150, 3.79249348248050154554852468057, 4.81121049574883958779355096319, 5.47215373239815982606467424212, 5.81835871848670283166700104716, 6.62092155118167853349415996974, 7.11072838228820363744423359939, 8.162887833307357061119312635822