| L(s) = 1 | − 2-s − 2.68·3-s + 4-s + 2.68·6-s + 4.59·7-s − 8-s + 4.22·9-s + 5.13·11-s − 2.68·12-s + 1.22·13-s − 4.59·14-s + 16-s + 4.68·17-s − 4.22·18-s − 4.59·19-s − 12.3·21-s − 5.13·22-s + 23-s + 2.68·24-s − 1.22·26-s − 3.28·27-s + 4.59·28-s + 3.37·29-s − 0.777·31-s − 32-s − 13.7·33-s − 4.68·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.55·3-s + 0.5·4-s + 1.09·6-s + 1.73·7-s − 0.353·8-s + 1.40·9-s + 1.54·11-s − 0.775·12-s + 0.338·13-s − 1.22·14-s + 0.250·16-s + 1.13·17-s − 0.995·18-s − 1.05·19-s − 2.69·21-s − 1.09·22-s + 0.208·23-s + 0.548·24-s − 0.239·26-s − 0.632·27-s + 0.868·28-s + 0.626·29-s − 0.139·31-s − 0.176·32-s − 2.40·33-s − 0.803·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9658966876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9658966876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 - 8.93T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−10.12048227074422448585924089022, −8.832790662230635775275021571195, −8.297421217678608371078629272566, −7.17446505293404682935345569038, −6.52023634843169657001551212315, −5.60229015874661185041588346918, −4.85699245367611369076996366602, −3.85209003540653046182139281946, −1.76661072126911022167516583972, −0.988449079867128240513664410925,
0.988449079867128240513664410925, 1.76661072126911022167516583972, 3.85209003540653046182139281946, 4.85699245367611369076996366602, 5.60229015874661185041588346918, 6.52023634843169657001551212315, 7.17446505293404682935345569038, 8.297421217678608371078629272566, 8.832790662230635775275021571195, 10.12048227074422448585924089022
