L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s + 3·7-s − 2·9-s + 8·10-s − 3·11-s − 2·12-s + 4·13-s − 6·14-s + 4·15-s − 4·16-s − 2·17-s + 4·18-s − 8·20-s − 3·21-s + 6·22-s + 4·23-s + 11·25-s − 8·26-s + 5·27-s + 6·28-s − 8·30-s + 31-s + 8·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s + 1.13·7-s − 2/3·9-s + 2.52·10-s − 0.904·11-s − 0.577·12-s + 1.10·13-s − 1.60·14-s + 1.03·15-s − 16-s − 0.485·17-s + 0.942·18-s − 1.78·20-s − 0.654·21-s + 1.27·22-s + 0.834·23-s + 11/5·25-s − 1.56·26-s + 0.962·27-s + 1.13·28-s − 1.46·30-s + 0.179·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.945355910979209686182413831237, −8.624784847131579311333753302442, −7.75264259175123186024886386744, −7.52277241263090313416301188741, −6.25709792192482309041805863604, −4.97531776109572988323043518052, −4.26171274569399943355761298605, −2.85480127665183279093650283922, −1.14714747189337797537655581095, 0,
1.14714747189337797537655581095, 2.85480127665183279093650283922, 4.26171274569399943355761298605, 4.97531776109572988323043518052, 6.25709792192482309041805863604, 7.52277241263090313416301188741, 7.75264259175123186024886386744, 8.624784847131579311333753302442, 8.945355910979209686182413831237