| L(s) = 1 | − 2-s + 1.25·3-s + 4-s − 2.03·5-s − 1.25·6-s − 2.12·7-s − 8-s − 1.41·9-s + 2.03·10-s + 4.10·11-s + 1.25·12-s − 4.74·13-s + 2.12·14-s − 2.56·15-s + 16-s + 6.58·17-s + 1.41·18-s − 8.27·19-s − 2.03·20-s − 2.67·21-s − 4.10·22-s − 4.52·23-s − 1.25·24-s − 0.848·25-s + 4.74·26-s − 5.55·27-s − 2.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.726·3-s + 0.5·4-s − 0.911·5-s − 0.513·6-s − 0.804·7-s − 0.353·8-s − 0.472·9-s + 0.644·10-s + 1.23·11-s + 0.363·12-s − 1.31·13-s + 0.568·14-s − 0.661·15-s + 0.250·16-s + 1.59·17-s + 0.333·18-s − 1.89·19-s − 0.455·20-s − 0.584·21-s − 0.875·22-s − 0.942·23-s − 0.256·24-s − 0.169·25-s + 0.931·26-s − 1.06·27-s − 0.402·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8834032276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8834032276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 + 8.27T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 - 0.442T + 29T^{2} \) |
| 31 | \( 1 - 0.451T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 5.83T + 41T^{2} \) |
| 43 | \( 1 + 6.40T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 - 4.05T + 53T^{2} \) |
| 59 | \( 1 + 8.25T + 59T^{2} \) |
| 61 | \( 1 - 6.44T + 61T^{2} \) |
| 67 | \( 1 - 0.512T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 1.36T + 97T^{2} \) |
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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
−8.318673344042370633040825069511, −7.912639664557360829550640524227, −7.23752619269181399300560018363, −6.38053209382212755494337949061, −5.75312819416044744120220309985, −4.30765982259631033668783745578, −3.75227482905181359984895505455, −2.87465287432770123205320560012, −2.07419462079727378642629311842, −0.54926600514210190885289254430,
0.54926600514210190885289254430, 2.07419462079727378642629311842, 2.87465287432770123205320560012, 3.75227482905181359984895505455, 4.30765982259631033668783745578, 5.75312819416044744120220309985, 6.38053209382212755494337949061, 7.23752619269181399300560018363, 7.912639664557360829550640524227, 8.318673344042370633040825069511
