| L(s) = 1 | + 0.554·2-s − 1.69·4-s + 2.80·5-s − 2.69·7-s − 2.04·8-s + 1.55·10-s + 1.19·11-s − 1.49·14-s + 2.24·16-s − 1.13·17-s + 1.93·19-s − 4.74·20-s + 0.664·22-s + 4.60·23-s + 2.85·25-s + 4.55·28-s + 7.89·29-s + 5.89·31-s + 5.34·32-s − 0.631·34-s − 7.54·35-s + 0.951·37-s + 1.07·38-s − 5.74·40-s − 3.31·41-s + 7.15·43-s − 2.02·44-s + ⋯ |
| L(s) = 1 | + 0.392·2-s − 0.846·4-s + 1.25·5-s − 1.01·7-s − 0.724·8-s + 0.491·10-s + 0.361·11-s − 0.399·14-s + 0.561·16-s − 0.275·17-s + 0.444·19-s − 1.06·20-s + 0.141·22-s + 0.959·23-s + 0.570·25-s + 0.860·28-s + 1.46·29-s + 1.05·31-s + 0.944·32-s − 0.108·34-s − 1.27·35-s + 0.156·37-s + 0.174·38-s − 0.907·40-s − 0.518·41-s + 1.09·43-s − 0.305·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.878324636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878324636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.554T + 2T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 0.951T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 7.15T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 5.87T + 53T^{2} \) |
| 59 | \( 1 - 0.0120T + 59T^{2} \) |
| 61 | \( 1 + 8.03T + 61T^{2} \) |
| 67 | \( 1 + 9.25T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 - 0.807T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 3.13T + 97T^{2} \) |
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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
−9.363006836300003868799215249364, −9.059444978076077215767312000989, −7.971426477260766816256698810758, −6.62616896668868059930454523370, −6.23029656461718535288941860176, −5.31894279368416713690989257950, −4.54039591717497767895511757678, −3.40259283597691604235609978299, −2.56638624975462701257920343514, −0.950211259332586604997058407720,
0.950211259332586604997058407720, 2.56638624975462701257920343514, 3.40259283597691604235609978299, 4.54039591717497767895511757678, 5.31894279368416713690989257950, 6.23029656461718535288941860176, 6.62616896668868059930454523370, 7.971426477260766816256698810758, 9.059444978076077215767312000989, 9.363006836300003868799215249364
