| L(s) = 1 | − 1.65·3-s + 5-s + 4.10·7-s − 0.251·9-s + 5.07·11-s − 3.93·13-s − 1.65·15-s − 1.21·17-s + 3.27·19-s − 6.80·21-s + 6.17·23-s + 25-s + 5.39·27-s + 2.73·29-s − 6.62·31-s − 8.41·33-s + 4.10·35-s + 0.852·37-s + 6.51·39-s − 4.07·41-s − 0.400·43-s − 0.251·45-s + 8.42·47-s + 9.85·49-s + 2.01·51-s − 12.9·53-s + 5.07·55-s + ⋯ |
| L(s) = 1 | − 0.957·3-s + 0.447·5-s + 1.55·7-s − 0.0837·9-s + 1.53·11-s − 1.09·13-s − 0.428·15-s − 0.294·17-s + 0.751·19-s − 1.48·21-s + 1.28·23-s + 0.200·25-s + 1.03·27-s + 0.507·29-s − 1.19·31-s − 1.46·33-s + 0.693·35-s + 0.140·37-s + 1.04·39-s − 0.636·41-s − 0.0611·43-s − 0.0374·45-s + 1.22·47-s + 1.40·49-s + 0.281·51-s − 1.77·53-s + 0.684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106801152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106801152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 - 0.852T + 37T^{2} \) |
| 41 | \( 1 + 4.07T + 41T^{2} \) |
| 43 | \( 1 + 0.400T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 - 5.83T + 73T^{2} \) |
| 79 | \( 1 - 9.07T + 79T^{2} \) |
| 83 | \( 1 - 5.00T + 83T^{2} \) |
| 89 | \( 1 + 5.51T + 89T^{2} \) |
| 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
−7.75022987361402759440847025889, −6.96160568165220674108648738566, −6.53163311737291204961876802235, −5.54797910352118445204674399502, −5.08971601242505773210327085895, −4.63039417922659069204849415417, −3.63461961480307737941067695302, −2.49166998362076215491005664980, −1.57858977975155996430787958615, −0.817805373758129911422087476186,
0.817805373758129911422087476186, 1.57858977975155996430787958615, 2.49166998362076215491005664980, 3.63461961480307737941067695302, 4.63039417922659069204849415417, 5.08971601242505773210327085895, 5.54797910352118445204674399502, 6.53163311737291204961876802235, 6.96160568165220674108648738566, 7.75022987361402759440847025889
