L(s) = 1 | − 2-s + 4-s + 5-s + 4·7-s − 8-s − 10-s + 2·11-s + 2·13-s − 4·14-s + 16-s + 2·17-s + 2·19-s + 20-s − 2·22-s + 25-s − 2·26-s + 4·28-s − 2·29-s + 4·31-s − 32-s − 2·34-s + 4·35-s + 37-s − 2·38-s − 40-s + 6·41-s − 4·43-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.458·19-s + 0.223·20-s − 0.426·22-s + 1/5·25-s − 0.392·26-s + 0.755·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s + 0.676·35-s + 0.164·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.063724825\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.063724825\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.748769502062719880669979484464, −7.85061419107578417511427026495, −7.44911319646635842053458117421, −6.38557362678220784025708982991, −5.71352219882885463253372405186, −4.86509043614248974467347621224, −3.97299573085911109082869407492, −2.79853495604275645593009425073, −1.71223534785965100948694097040, −1.07428846969378408368766263038,
1.07428846969378408368766263038, 1.71223534785965100948694097040, 2.79853495604275645593009425073, 3.97299573085911109082869407492, 4.86509043614248974467347621224, 5.71352219882885463253372405186, 6.38557362678220784025708982991, 7.44911319646635842053458117421, 7.85061419107578417511427026495, 8.748769502062719880669979484464