| L(s) = 1 | − 4.01·2-s + 3·3-s + 8.13·4-s + 0.375·5-s − 12.0·6-s − 23.3·7-s − 0.549·8-s + 9·9-s − 1.50·10-s + 33.7·11-s + 24.4·12-s − 47.8·13-s + 93.7·14-s + 1.12·15-s − 62.8·16-s + 123.·17-s − 36.1·18-s − 142.·19-s + 3.05·20-s − 70.0·21-s − 135.·22-s + 23·23-s − 1.64·24-s − 124.·25-s + 192.·26-s + 27·27-s − 189.·28-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.01·4-s + 0.0335·5-s − 0.819·6-s − 1.26·7-s − 0.0242·8-s + 0.333·9-s − 0.0477·10-s + 0.924·11-s + 0.587·12-s − 1.02·13-s + 1.79·14-s + 0.0193·15-s − 0.982·16-s + 1.75·17-s − 0.473·18-s − 1.71·19-s + 0.0341·20-s − 0.727·21-s − 1.31·22-s + 0.208·23-s − 0.0140·24-s − 0.998·25-s + 1.45·26-s + 0.192·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7831448700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7831448700\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 4.01T + 8T^{2} \) |
| 5 | \( 1 - 0.375T + 125T^{2} \) |
| 7 | \( 1 + 23.3T + 343T^{2} \) |
| 11 | \( 1 - 33.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 74.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 113.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 692.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 77.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 637.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 915.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 788.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.38e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 644.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.29e3T + 9.12e5T^{2} \) |
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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
−9.113952477945820418260014463906, −8.006852257826547369877195734495, −7.59474781616455053249145913076, −6.65231965544683658076682981618, −6.05819645145669336434654498170, −4.58101113917224968367628185759, −3.62857754154136745635512628223, −2.64205429728000711695804920632, −1.64375827088065493766525476088, −0.49373480727003821008191415654,
0.49373480727003821008191415654, 1.64375827088065493766525476088, 2.64205429728000711695804920632, 3.62857754154136745635512628223, 4.58101113917224968367628185759, 6.05819645145669336434654498170, 6.65231965544683658076682981618, 7.59474781616455053249145913076, 8.006852257826547369877195734495, 9.113952477945820418260014463906
