L(s) = 1 | + 3-s − 5-s − 3.72·7-s + 9-s − 4.64·11-s − 13-s − 15-s + 1.07·17-s − 3.72·19-s − 3.72·21-s − 4.92·23-s + 25-s + 27-s − 2.92·29-s − 2.79·31-s − 4.64·33-s + 3.72·35-s + 3.85·37-s − 39-s + 5.44·41-s + 8·43-s − 45-s − 12.0·47-s + 6.85·49-s + 1.07·51-s + 9.44·53-s + 4.64·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.40·7-s + 0.333·9-s − 1.40·11-s − 0.277·13-s − 0.258·15-s + 0.260·17-s − 0.853·19-s − 0.812·21-s − 1.02·23-s + 0.200·25-s + 0.192·27-s − 0.543·29-s − 0.502·31-s − 0.808·33-s + 0.629·35-s + 0.633·37-s − 0.160·39-s + 0.850·41-s + 1.21·43-s − 0.149·45-s − 1.76·47-s + 0.978·49-s + 0.150·51-s + 1.29·53-s + 0.626·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9770453747\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9770453747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 + 4.92T + 23T^{2} \) |
| 29 | \( 1 + 2.92T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 5.44T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 9.44T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 1.20T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 5.59T + 79T^{2} \) |
| 83 | \( 1 - 6.49T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 4.77T + 97T^{2} \) |
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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
−8.003513678073130281318524875112, −7.46534072715145828564545018310, −6.71425850306972327768559039411, −5.97174117475300137076924948591, −5.22550037884518891650079345469, −4.18600037263721053223284673551, −3.60162315989631273841658635288, −2.75265185643774996358621755578, −2.16961811285959328403140922843, −0.46529919784726677985562126642,
0.46529919784726677985562126642, 2.16961811285959328403140922843, 2.75265185643774996358621755578, 3.60162315989631273841658635288, 4.18600037263721053223284673551, 5.22550037884518891650079345469, 5.97174117475300137076924948591, 6.71425850306972327768559039411, 7.46534072715145828564545018310, 8.003513678073130281318524875112