| L(s) = 1 | − 3.12·3-s − 1.51·7-s + 6.76·9-s − 4.24·11-s − 4.15·13-s + 3.51·17-s + 19-s + 4.73·21-s − 8.73·23-s − 11.7·27-s + 1.45·29-s + 4.96·31-s + 13.2·33-s − 7.60·37-s + 12.9·39-s − 9.21·41-s − 8.31·43-s + 5.28·47-s − 4.70·49-s − 10.9·51-s − 0.155·53-s − 3.12·57-s + 2.48·59-s − 4.49·61-s − 10.2·63-s + 7.43·67-s + 27.2·69-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 0.572·7-s + 2.25·9-s − 1.28·11-s − 1.15·13-s + 0.852·17-s + 0.229·19-s + 1.03·21-s − 1.82·23-s − 2.26·27-s + 0.270·29-s + 0.892·31-s + 2.31·33-s − 1.25·37-s + 2.07·39-s − 1.43·41-s − 1.26·43-s + 0.770·47-s − 0.672·49-s − 1.53·51-s − 0.0213·53-s − 0.413·57-s + 0.323·59-s − 0.576·61-s − 1.29·63-s + 0.908·67-s + 3.28·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1701276880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1701276880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 7 | \( 1 + 1.51T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 23 | \( 1 + 8.73T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 - 4.96T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 + 9.21T + 41T^{2} \) |
| 43 | \( 1 + 8.31T + 43T^{2} \) |
| 47 | \( 1 - 5.28T + 47T^{2} \) |
| 53 | \( 1 + 0.155T + 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 + 4.49T + 61T^{2} \) |
| 67 | \( 1 - 7.43T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 - 0.310T + 79T^{2} \) |
| 83 | \( 1 + 8.96T + 83T^{2} \) |
| 89 | \( 1 - 0.719T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.75913446090789421415057135330, −6.93254801686219919125308978557, −6.50130876489893173906471838194, −5.58731816751001869143043698732, −5.28466237274120067411903446255, −4.63413288078045571317512414255, −3.68086935884882242242708462923, −2.64666835717095979727449375642, −1.55627146256674749058188853425, −0.22572363551697042112645835479,
0.22572363551697042112645835479, 1.55627146256674749058188853425, 2.64666835717095979727449375642, 3.68086935884882242242708462923, 4.63413288078045571317512414255, 5.28466237274120067411903446255, 5.58731816751001869143043698732, 6.50130876489893173906471838194, 6.93254801686219919125308978557, 7.75913446090789421415057135330
