| L(s) = 1 | + 1.01·2-s − 1.05·3-s − 0.974·4-s + 3.24·5-s − 1.06·6-s − 1.79·7-s − 3.01·8-s − 1.89·9-s + 3.28·10-s + 6.09·11-s + 1.02·12-s − 1.81·14-s − 3.41·15-s − 1.09·16-s − 4.79·17-s − 1.91·18-s + 1.84·19-s − 3.16·20-s + 1.88·21-s + 6.17·22-s + 7.94·23-s + 3.16·24-s + 5.52·25-s + 5.14·27-s + 1.75·28-s − 2.23·29-s − 3.45·30-s + ⋯ |
| L(s) = 1 | + 0.715·2-s − 0.607·3-s − 0.487·4-s + 1.45·5-s − 0.434·6-s − 0.678·7-s − 1.06·8-s − 0.631·9-s + 1.03·10-s + 1.83·11-s + 0.295·12-s − 0.485·14-s − 0.880·15-s − 0.274·16-s − 1.16·17-s − 0.452·18-s + 0.424·19-s − 0.707·20-s + 0.411·21-s + 1.31·22-s + 1.65·23-s + 0.646·24-s + 1.10·25-s + 0.990·27-s + 0.330·28-s − 0.414·29-s − 0.630·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169545814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169545814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.01T + 2T^{2} \) |
| 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 1.79T + 7T^{2} \) |
| 11 | \( 1 - 6.09T + 11T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 - 7.94T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 - 0.0498T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4.00T + 47T^{2} \) |
| 53 | \( 1 - 0.588T + 53T^{2} \) |
| 59 | \( 1 + 3.49T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 1.72T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.596804964147975924260055190699, −6.87558057907506369747959215941, −6.58651423993100587658640896706, −5.98077235658734072714778356070, −5.31511426202785625036114182034, −4.76406078206073713495335627531, −3.70998278182191762967743596725, −3.05282962982674690380167877029, −1.94221127702378238774114081970, −0.74429277773540886873715853865,
0.74429277773540886873715853865, 1.94221127702378238774114081970, 3.05282962982674690380167877029, 3.70998278182191762967743596725, 4.76406078206073713495335627531, 5.31511426202785625036114182034, 5.98077235658734072714778356070, 6.58651423993100587658640896706, 6.87558057907506369747959215941, 8.596804964147975924260055190699
