| L(s) = 1 | + 4.74·3-s + 20.7·5-s + 3.45·7-s − 4.45·9-s + 1.02·11-s − 34.4·13-s + 98.6·15-s − 51.2·17-s + 141.·19-s + 16.4·21-s − 79.2·23-s + 306.·25-s − 149.·27-s + 19.9·29-s + 31·31-s + 4.88·33-s + 71.7·35-s − 300.·37-s − 163.·39-s + 41.7·41-s + 142.·43-s − 92.4·45-s + 275.·47-s − 331.·49-s − 243.·51-s − 122.·53-s + 21.3·55-s + ⋯ |
| L(s) = 1 | + 0.913·3-s + 1.85·5-s + 0.186·7-s − 0.164·9-s + 0.0281·11-s − 0.735·13-s + 1.69·15-s − 0.731·17-s + 1.70·19-s + 0.170·21-s − 0.718·23-s + 2.45·25-s − 1.06·27-s + 0.127·29-s + 0.179·31-s + 0.0257·33-s + 0.346·35-s − 1.33·37-s − 0.671·39-s + 0.159·41-s + 0.506·43-s − 0.306·45-s + 0.854·47-s − 0.965·49-s − 0.668·51-s − 0.318·53-s + 0.0523·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.689282949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.689282949\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 - 31T \) |
| good | 3 | \( 1 - 4.74T + 27T^{2} \) |
| 5 | \( 1 - 20.7T + 125T^{2} \) |
| 7 | \( 1 - 3.45T + 343T^{2} \) |
| 11 | \( 1 - 1.02T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 19.9T + 2.43e4T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 41.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 702.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 152.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 209.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 885.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.12e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 74.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 941.T + 9.12e5T^{2} \) |
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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
−13.35499175801067085339452062756, −12.05184660756314228171218904996, −10.56030128918214930958252606911, −9.543509341395848069598271160630, −8.994029020048615499134817655203, −7.58356595146958725443623787670, −6.16215009351131415887146780040, −5.05982355013745525041446132072, −2.95847655538707576171147366116, −1.84487722381649790553114215227,
1.84487722381649790553114215227, 2.95847655538707576171147366116, 5.05982355013745525041446132072, 6.16215009351131415887146780040, 7.58356595146958725443623787670, 8.994029020048615499134817655203, 9.543509341395848069598271160630, 10.56030128918214930958252606911, 12.05184660756314228171218904996, 13.35499175801067085339452062756
