| L(s) = 1 | − 2·2-s + 5.90·3-s + 4·4-s − 11.8·6-s + 1.63·7-s − 8·8-s + 7.83·9-s + 42.3·11-s + 23.6·12-s + 18.1·13-s − 3.27·14-s + 16·16-s + 122.·17-s − 15.6·18-s + 87.0·19-s + 9.65·21-s − 84.7·22-s + 23·23-s − 47.2·24-s − 36.3·26-s − 113.·27-s + 6.54·28-s − 187.·29-s + 167.·31-s − 32·32-s + 250.·33-s − 245.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.13·3-s + 0.5·4-s − 0.803·6-s + 0.0882·7-s − 0.353·8-s + 0.290·9-s + 1.16·11-s + 0.567·12-s + 0.388·13-s − 0.0624·14-s + 0.250·16-s + 1.74·17-s − 0.205·18-s + 1.05·19-s + 0.100·21-s − 0.821·22-s + 0.208·23-s − 0.401·24-s − 0.274·26-s − 0.806·27-s + 0.0441·28-s − 1.20·29-s + 0.971·31-s − 0.176·32-s + 1.31·33-s − 1.23·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.769951355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.769951355\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 5.90T + 27T^{2} \) |
| 7 | \( 1 - 1.63T + 343T^{2} \) |
| 11 | \( 1 - 42.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 87.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 187.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 245.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 52.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 669.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 425.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 74.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 937.T + 9.12e5T^{2} \) |
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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
−9.337378294730320996310005278174, −8.691464410625558127209174845151, −7.88354649623682873792292918770, −7.31500801282113761254218562857, −6.23660777955378040965122654289, −5.24910266914726544051843555033, −3.66211598251460802698794717116, −3.23488961858563785820968652259, −1.89881370030052494107551114612, −0.966067361723845895074759816554,
0.966067361723845895074759816554, 1.89881370030052494107551114612, 3.23488961858563785820968652259, 3.66211598251460802698794717116, 5.24910266914726544051843555033, 6.23660777955378040965122654289, 7.31500801282113761254218562857, 7.88354649623682873792292918770, 8.691464410625558127209174845151, 9.337378294730320996310005278174
