L(s) = 1 | − 2.50·3-s − 3.40·5-s + 7-s + 3.27·9-s − 11-s − 13-s + 8.51·15-s + 4.74·17-s + 6.83·19-s − 2.50·21-s + 4.61·23-s + 6.56·25-s − 0.683·27-s + 5.06·29-s + 3.54·31-s + 2.50·33-s − 3.40·35-s − 6.49·37-s + 2.50·39-s − 1.94·41-s − 3.11·43-s − 11.1·45-s + 3.72·47-s + 49-s − 11.8·51-s + 2.20·53-s + 3.40·55-s + ⋯ |
L(s) = 1 | − 1.44·3-s − 1.52·5-s + 0.377·7-s + 1.09·9-s − 0.301·11-s − 0.277·13-s + 2.19·15-s + 1.15·17-s + 1.56·19-s − 0.546·21-s + 0.961·23-s + 1.31·25-s − 0.131·27-s + 0.940·29-s + 0.637·31-s + 0.435·33-s − 0.574·35-s − 1.06·37-s + 0.401·39-s − 0.303·41-s − 0.475·43-s − 1.65·45-s + 0.543·47-s + 0.142·49-s − 1.66·51-s + 0.302·53-s + 0.458·55-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
\(L(1)\) |
\(\approx\) |
\(0.8290095593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8290095593\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.50T + 3T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 - 4.61T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 + 1.94T + 41T^{2} \) |
| 43 | \( 1 + 3.11T + 43T^{2} \) |
| 47 | \( 1 - 3.72T + 47T^{2} \) |
| 53 | \( 1 - 2.20T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 + 9.11T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 - 2.08T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 - 3.09T + 97T^{2} \) |
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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
−7.58266739228911749326727683971, −7.28163613310868988473367674172, −6.48078978666574772933507918963, −5.58080939104711471869376750798, −5.00543168672806217565459923511, −4.59536106745626822406642506857, −3.54503679265529292431471970059, −2.94685296956336963033932472468, −1.25935154167695522133222100457, −0.56642332168074100064339288061,
0.56642332168074100064339288061, 1.25935154167695522133222100457, 2.94685296956336963033932472468, 3.54503679265529292431471970059, 4.59536106745626822406642506857, 5.00543168672806217565459923511, 5.58080939104711471869376750798, 6.48078978666574772933507918963, 7.28163613310868988473367674172, 7.58266739228911749326727683971