L(s) = 1 | + 3.25·3-s − 0.215·5-s + 7-s + 7.56·9-s − 11-s − 13-s − 0.701·15-s − 1.35·17-s + 5.10·19-s + 3.25·21-s + 4.33·23-s − 4.95·25-s + 14.8·27-s − 0.702·29-s + 9.03·31-s − 3.25·33-s − 0.215·35-s − 2.18·37-s − 3.25·39-s + 3.79·41-s + 2.91·43-s − 1.63·45-s + 10.5·47-s + 49-s − 4.40·51-s − 8.94·53-s + 0.215·55-s + ⋯ |
L(s) = 1 | + 1.87·3-s − 0.0964·5-s + 0.377·7-s + 2.52·9-s − 0.301·11-s − 0.277·13-s − 0.181·15-s − 0.328·17-s + 1.17·19-s + 0.709·21-s + 0.903·23-s − 0.990·25-s + 2.85·27-s − 0.130·29-s + 1.62·31-s − 0.565·33-s − 0.0364·35-s − 0.359·37-s − 0.520·39-s + 0.593·41-s + 0.444·43-s − 0.243·45-s + 1.54·47-s + 0.142·49-s − 0.616·51-s − 1.22·53-s + 0.0290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.836108541\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.836108541\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 + 0.215T + 5T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 + 0.702T + 29T^{2} \) |
| 31 | \( 1 - 9.03T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−7.80401125873373751925659653267, −7.49342213964815289267456376996, −6.76734113468614518516780650562, −5.71617498899189024550160309427, −4.71760898486404074812670148238, −4.22385966540770329732155044031, −3.27270323185389540936114166455, −2.77861199621897909463387438681, −1.99091529368464379479966126902, −1.05192609459721046642260147867,
1.05192609459721046642260147867, 1.99091529368464379479966126902, 2.77861199621897909463387438681, 3.27270323185389540936114166455, 4.22385966540770329732155044031, 4.71760898486404074812670148238, 5.71617498899189024550160309427, 6.76734113468614518516780650562, 7.49342213964815289267456376996, 7.80401125873373751925659653267