L(s) = 1 | − 1.70·3-s − 2.63·7-s − 0.0783·9-s + 5.41·11-s − 6.34·13-s + 3.41·17-s + 3.26·19-s + 4.49·21-s − 1.36·23-s + 5.26·27-s − 2·29-s + 4.68·31-s − 9.26·33-s + 5.75·37-s + 10.8·39-s − 7.75·41-s + 4.44·43-s − 4.78·47-s − 0.0783·49-s − 5.84·51-s − 1.65·53-s − 5.57·57-s − 3.26·59-s + 2.49·61-s + 0.206·63-s + 7.86·67-s + 2.34·69-s + ⋯ |
L(s) = 1 | − 0.986·3-s − 0.994·7-s − 0.0261·9-s + 1.63·11-s − 1.75·13-s + 0.829·17-s + 0.748·19-s + 0.981·21-s − 0.285·23-s + 1.01·27-s − 0.371·29-s + 0.840·31-s − 1.61·33-s + 0.946·37-s + 1.73·39-s − 1.21·41-s + 0.678·43-s − 0.698·47-s − 0.0111·49-s − 0.818·51-s − 0.227·53-s − 0.738·57-s − 0.424·59-s + 0.319·61-s + 0.0259·63-s + 0.960·67-s + 0.281·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 2.63T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 - 4.44T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 - 2.49T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.52T + 89T^{2} \) |
| 97 | \( 1 + 4.58T + 97T^{2} \) |
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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
−8.258667860640664203230314910715, −7.29719028093854592966903450231, −6.68299208929099535511810146973, −6.08587757693125828610539297517, −5.32234051715402591864990752976, −4.53456042811516392832498522765, −3.52893386632641623073122104097, −2.66936833990450223717936013331, −1.19537397572838742088773266482, 0,
1.19537397572838742088773266482, 2.66936833990450223717936013331, 3.52893386632641623073122104097, 4.53456042811516392832498522765, 5.32234051715402591864990752976, 6.08587757693125828610539297517, 6.68299208929099535511810146973, 7.29719028093854592966903450231, 8.258667860640664203230314910715