| L(s) = 1 | − 3.12·3-s + 5-s + 1.51·7-s + 6.76·9-s + 4.24·11-s − 4.15·13-s − 3.12·15-s − 3.51·17-s − 19-s − 4.73·21-s + 8.73·23-s + 25-s − 11.7·27-s − 1.45·29-s + 4.96·31-s − 13.2·33-s + 1.51·35-s − 7.60·37-s + 12.9·39-s − 9.21·41-s − 8.31·43-s + 6.76·45-s − 5.28·47-s − 4.70·49-s + 10.9·51-s − 0.155·53-s + 4.24·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 0.447·5-s + 0.572·7-s + 2.25·9-s + 1.28·11-s − 1.15·13-s − 0.806·15-s − 0.852·17-s − 0.229·19-s − 1.03·21-s + 1.82·23-s + 0.200·25-s − 2.26·27-s − 0.270·29-s + 0.892·31-s − 2.31·33-s + 0.256·35-s − 1.25·37-s + 2.07·39-s − 1.43·41-s − 1.26·43-s + 1.00·45-s − 0.770·47-s − 0.672·49-s + 1.53·51-s − 0.0213·53-s + 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 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
−7.30080376077402126920512200752, −6.77683606269955075697780742848, −6.43657434991135070680741895311, −5.48262033806665469354359955986, −4.79418331645031123147628770715, −4.60620831537287531706031345170, −3.33432258732142566730945570160, −1.92480070103248348434855175060, −1.22028073388181966980160800317, 0,
1.22028073388181966980160800317, 1.92480070103248348434855175060, 3.33432258732142566730945570160, 4.60620831537287531706031345170, 4.79418331645031123147628770715, 5.48262033806665469354359955986, 6.43657434991135070680741895311, 6.77683606269955075697780742848, 7.30080376077402126920512200752
