| L(s) = 1 | + 2-s + 2.48·3-s + 4-s + 3.25·5-s + 2.48·6-s + 4.41·7-s + 8-s + 3.17·9-s + 3.25·10-s − 11-s + 2.48·12-s − 2.96·13-s + 4.41·14-s + 8.09·15-s + 16-s − 6.19·17-s + 3.17·18-s + 0.0372·19-s + 3.25·20-s + 10.9·21-s − 22-s − 6.66·23-s + 2.48·24-s + 5.60·25-s − 2.96·26-s + 0.430·27-s + 4.41·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.43·3-s + 0.5·4-s + 1.45·5-s + 1.01·6-s + 1.67·7-s + 0.353·8-s + 1.05·9-s + 1.02·10-s − 0.301·11-s + 0.717·12-s − 0.823·13-s + 1.18·14-s + 2.08·15-s + 0.250·16-s − 1.50·17-s + 0.747·18-s + 0.00855·19-s + 0.728·20-s + 2.39·21-s − 0.213·22-s − 1.38·23-s + 0.507·24-s + 1.12·25-s − 0.582·26-s + 0.0829·27-s + 0.835·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.376603564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.376603564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 - 0.0372T + 19T^{2} \) |
| 23 | \( 1 + 6.66T + 23T^{2} \) |
| 29 | \( 1 - 5.72T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 - 8.80T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 5.21T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 7.66T + 71T^{2} \) |
| 73 | \( 1 - 5.80T + 73T^{2} \) |
| 79 | \( 1 + 0.538T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 6.43T + 97T^{2} \) |
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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
−8.526932056937039855671303603502, −7.64993228323236088864471675313, −7.04387282487826566122948631888, −6.03660189558731238756787693125, −5.27061908714873897508103948477, −4.62079457176257397191164009741, −3.87682131636064879338988360061, −2.46629218710106488124609475109, −2.30789352101824997681960652339, −1.59582994578002664309708491597,
1.59582994578002664309708491597, 2.30789352101824997681960652339, 2.46629218710106488124609475109, 3.87682131636064879338988360061, 4.62079457176257397191164009741, 5.27061908714873897508103948477, 6.03660189558731238756787693125, 7.04387282487826566122948631888, 7.64993228323236088864471675313, 8.526932056937039855671303603502
