| L(s) = 1 | − 9.38·3-s − 9.38·5-s + 61·9-s − 20·11-s − 65.6·13-s + 88·15-s − 56.2·17-s + 9.38·19-s − 48·23-s − 37·25-s − 318.·27-s − 166·29-s − 206.·31-s + 187.·33-s − 78·37-s + 616·39-s − 393.·41-s − 436·43-s − 572.·45-s + 206.·47-s + 528·51-s + 62·53-s + 187.·55-s − 88·57-s − 666.·59-s − 272.·61-s + 616·65-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 0.839·5-s + 2.25·9-s − 0.548·11-s − 1.40·13-s + 1.51·15-s − 0.803·17-s + 0.113·19-s − 0.435·23-s − 0.295·25-s − 2.27·27-s − 1.06·29-s − 1.19·31-s + 0.989·33-s − 0.346·37-s + 2.52·39-s − 1.50·41-s − 1.54·43-s − 1.89·45-s + 0.640·47-s + 1.44·51-s + 0.160·53-s + 0.459·55-s − 0.204·57-s − 1.46·59-s − 0.571·61-s + 1.17·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05149867083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05149867083\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 9.38T + 27T^{2} \) |
| 5 | \( 1 + 9.38T + 125T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48T + 1.21e4T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 436T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62T + 1.48e5T^{2} \) |
| 59 | \( 1 + 666.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 272.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 580T + 3.00e5T^{2} \) |
| 71 | \( 1 - 544T + 3.57e5T^{2} \) |
| 73 | \( 1 - 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 680T + 4.93e5T^{2} \) |
| 83 | \( 1 - 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.50e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 656.T + 9.12e5T^{2} \) |
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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
−10.21152593253292338265482546199, −9.268430322961435867646996430079, −7.82871505865705844366378400183, −7.22876684816720034287515534679, −6.37980088823679526490750228348, −5.29405634170247363279822575647, −4.78698730697163946558385710903, −3.70595883534113406332860892615, −1.91113353195262413988467369754, −0.13363686003472502454479406253,
0.13363686003472502454479406253, 1.91113353195262413988467369754, 3.70595883534113406332860892615, 4.78698730697163946558385710903, 5.29405634170247363279822575647, 6.37980088823679526490750228348, 7.22876684816720034287515534679, 7.82871505865705844366378400183, 9.268430322961435867646996430079, 10.21152593253292338265482546199
