| L(s) = 1 | + 3·3-s − 5·5-s − 9.18·7-s + 9·9-s + 8.91·11-s + 13·13-s − 15·15-s + 8.31·17-s − 71.9·19-s − 27.5·21-s + 14.3·23-s + 25·25-s + 27·27-s − 123.·29-s + 253.·31-s + 26.7·33-s + 45.9·35-s − 57.9·37-s + 39·39-s + 130.·41-s − 73.9·43-s − 45·45-s − 48.1·47-s − 258.·49-s + 24.9·51-s + 327.·53-s − 44.5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.495·7-s + 0.333·9-s + 0.244·11-s + 0.277·13-s − 0.258·15-s + 0.118·17-s − 0.868·19-s − 0.286·21-s + 0.129·23-s + 0.200·25-s + 0.192·27-s − 0.791·29-s + 1.46·31-s + 0.141·33-s + 0.221·35-s − 0.257·37-s + 0.160·39-s + 0.495·41-s − 0.262·43-s − 0.149·45-s − 0.149·47-s − 0.754·49-s + 0.0684·51-s + 0.848·53-s − 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.167155702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167155702\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 + 9.18T + 343T^{2} \) |
| 11 | \( 1 - 8.91T + 1.33e3T^{2} \) |
| 17 | \( 1 - 8.31T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 253.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 73.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 48.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 327.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 338.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 558.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 592.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 704.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 948.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 534.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 127.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.46e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−8.915681816232468203262053744075, −8.396377101103834478203680133881, −7.51986330316175673709566589029, −6.72654437056104894289936380277, −5.94491245981687299934001878803, −4.72540020001067192108062893023, −3.88910994678453111537513885684, −3.09771860579574954849915288498, −2.02604339203602168627322527497, −0.68711048288155108778909403881,
0.68711048288155108778909403881, 2.02604339203602168627322527497, 3.09771860579574954849915288498, 3.88910994678453111537513885684, 4.72540020001067192108062893023, 5.94491245981687299934001878803, 6.72654437056104894289936380277, 7.51986330316175673709566589029, 8.396377101103834478203680133881, 8.915681816232468203262053744075
