L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 3·5-s + 4·6-s − 5·7-s + 9-s + 6·10-s − 4·11-s − 4·12-s − 3·13-s + 10·14-s + 6·15-s − 4·16-s − 6·17-s − 2·18-s + 5·19-s − 6·20-s + 10·21-s + 8·22-s − 4·23-s + 4·25-s + 6·26-s + 4·27-s − 10·28-s − 6·29-s − 12·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.34·5-s + 1.63·6-s − 1.88·7-s + 1/3·9-s + 1.89·10-s − 1.20·11-s − 1.15·12-s − 0.832·13-s + 2.67·14-s + 1.54·15-s − 16-s − 1.45·17-s − 0.471·18-s + 1.14·19-s − 1.34·20-s + 2.18·21-s + 1.70·22-s − 0.834·23-s + 4/5·25-s + 1.17·26-s + 0.769·27-s − 1.88·28-s − 1.11·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 389 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−10.35143331288149667136655977862, −9.633078802184913454726670089801, −8.633205244563326241602650564321, −7.47490749578543088938020099445, −6.98596665282868921801197833011, −5.79340263392836527147559896903, −4.41689608366525782922561294304, −2.87609907126046520176342609472, 0, 0,
2.87609907126046520176342609472, 4.41689608366525782922561294304, 5.79340263392836527147559896903, 6.98596665282868921801197833011, 7.47490749578543088938020099445, 8.633205244563326241602650564321, 9.633078802184913454726670089801, 10.35143331288149667136655977862