| L(s) = 1 | − 3.34·3-s + 2.46·5-s − 2.50·7-s + 8.16·9-s + 2.72·11-s − 4.45·13-s − 8.24·15-s + 1.47·17-s + 4.54·19-s + 8.35·21-s − 3.69·23-s + 1.08·25-s − 17.2·27-s − 4.52·29-s + 1.62·31-s − 9.09·33-s − 6.16·35-s + 10.8·37-s + 14.8·39-s − 2.86·41-s − 3.14·43-s + 20.1·45-s − 1.65·47-s − 0.749·49-s − 4.93·51-s + 12.3·53-s + 6.71·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 1.10·5-s − 0.944·7-s + 2.72·9-s + 0.820·11-s − 1.23·13-s − 2.12·15-s + 0.358·17-s + 1.04·19-s + 1.82·21-s − 0.769·23-s + 0.216·25-s − 3.32·27-s − 0.841·29-s + 0.291·31-s − 1.58·33-s − 1.04·35-s + 1.78·37-s + 2.38·39-s − 0.447·41-s − 0.480·43-s + 3.00·45-s − 0.241·47-s − 0.107·49-s − 0.690·51-s + 1.69·53-s + 0.905·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8909956422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8909956422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 2.86T + 41T^{2} \) |
| 43 | \( 1 + 3.14T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 9.60T + 71T^{2} \) |
| 73 | \( 1 - 0.0933T + 73T^{2} \) |
| 79 | \( 1 - 6.20T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−9.868142976129538007292865624972, −9.394320344692584857172365142382, −7.61900083144535194940120647361, −6.81239174632004247834945931546, −6.20052903303924753031601029362, −5.59421765211002567922725812794, −4.87821632258351272157908962020, −3.73408838425205235016164032486, −2.08560275905975347737519616713, −0.76114014011926294330767719391,
0.76114014011926294330767719391, 2.08560275905975347737519616713, 3.73408838425205235016164032486, 4.87821632258351272157908962020, 5.59421765211002567922725812794, 6.20052903303924753031601029362, 6.81239174632004247834945931546, 7.61900083144535194940120647361, 9.394320344692584857172365142382, 9.868142976129538007292865624972
