| L(s) = 1 | + 2-s + 4-s − 1.76·5-s + 8-s − 1.76·10-s + 6.12·11-s − 0.760·13-s + 16-s − 6.84·17-s + 1.94·19-s − 1.76·20-s + 6.12·22-s − 0.421·23-s − 1.89·25-s − 0.760·26-s − 1.46·29-s − 7.70·31-s + 32-s − 6.84·34-s − 2.88·37-s + 1.94·38-s − 1.76·40-s − 6.94·41-s − 8.66·43-s + 6.12·44-s − 0.421·46-s − 1.66·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.787·5-s + 0.353·8-s − 0.556·10-s + 1.84·11-s − 0.211·13-s + 0.250·16-s − 1.65·17-s + 0.445·19-s − 0.393·20-s + 1.30·22-s − 0.0877·23-s − 0.379·25-s − 0.149·26-s − 0.271·29-s − 1.38·31-s + 0.176·32-s − 1.17·34-s − 0.474·37-s + 0.315·38-s − 0.278·40-s − 1.08·41-s − 1.32·43-s + 0.923·44-s − 0.0620·46-s − 0.242·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.76T + 5T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + 0.760T + 13T^{2} \) |
| 17 | \( 1 + 6.84T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 0.421T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 + 2.88T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 + 1.98T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.306T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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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
−7.16887213449714497488629000064, −6.85317344050322828518476338189, −6.21278527564529166064887957910, −5.26724757173210650754298956763, −4.56280973880038910559521995565, −3.76344554664785058407217297356, −3.56892389892465679867061488907, −2.24764112835169548083672397649, −1.46111242121715447322044230913, 0,
1.46111242121715447322044230913, 2.24764112835169548083672397649, 3.56892389892465679867061488907, 3.76344554664785058407217297356, 4.56280973880038910559521995565, 5.26724757173210650754298956763, 6.21278527564529166064887957910, 6.85317344050322828518476338189, 7.16887213449714497488629000064
