| L(s) = 1 | − 3.41·5-s − 0.291·7-s + 6.12·11-s + 2.59·13-s + 3.77·17-s − 1.94·19-s + 4.34·23-s + 6.64·25-s − 1.08·29-s + 4.46·31-s + 0.993·35-s − 6.62·37-s − 4.23·41-s − 4.37·43-s − 1.99·47-s − 6.91·49-s + 12.6·53-s − 20.9·55-s − 13.9·59-s + 11.3·61-s − 8.86·65-s − 5.07·67-s + 9.79·71-s − 4.08·73-s − 1.78·77-s − 1.63·79-s + 10.9·83-s + ⋯ |
| L(s) = 1 | − 1.52·5-s − 0.110·7-s + 1.84·11-s + 0.720·13-s + 0.915·17-s − 0.447·19-s + 0.905·23-s + 1.32·25-s − 0.202·29-s + 0.801·31-s + 0.167·35-s − 1.08·37-s − 0.661·41-s − 0.666·43-s − 0.290·47-s − 0.987·49-s + 1.73·53-s − 2.81·55-s − 1.81·59-s + 1.45·61-s − 1.09·65-s − 0.620·67-s + 1.16·71-s − 0.477·73-s − 0.203·77-s − 0.184·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655344343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655344343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 0.291T + 7T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 - 4.34T + 23T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 + 1.99T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 5.07T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 4.08T + 73T^{2} \) |
| 79 | \( 1 + 1.63T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 6.60T + 89T^{2} \) |
| 97 | \( 1 - 0.569T + 97T^{2} \) |
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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
−8.233687951029921949093973529822, −7.27972329421656491453587297079, −6.78836374042568942383816144619, −6.12829787928670552896016717531, −5.05331219828982281455337254741, −4.25807143835659626623102676415, −3.62046310999008611855609005398, −3.20060432442170203471101112820, −1.61436331367182080386592478726, −0.72258393863504163716895506320,
0.72258393863504163716895506320, 1.61436331367182080386592478726, 3.20060432442170203471101112820, 3.62046310999008611855609005398, 4.25807143835659626623102676415, 5.05331219828982281455337254741, 6.12829787928670552896016717531, 6.78836374042568942383816144619, 7.27972329421656491453587297079, 8.233687951029921949093973529822
