| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 11-s + 12-s − 2·13-s + 2·15-s + 16-s + 6·17-s − 18-s + 4·19-s + 2·20-s − 22-s − 4·23-s − 24-s − 25-s + 2·26-s + 27-s + 2·29-s − 2·30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.154636001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154636001\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.736150936012440857648978735320, −7.84439006039971069748622971041, −7.49024027339887201227457052610, −6.43679449975526542026435018487, −5.79323544635467042022952173625, −4.93054069578340184917711747999, −3.71446826599373971796272627437, −2.83711647406054313257956134718, −1.96943863395594925899870544120, −1.00454239585262764496288710387,
1.00454239585262764496288710387, 1.96943863395594925899870544120, 2.83711647406054313257956134718, 3.71446826599373971796272627437, 4.93054069578340184917711747999, 5.79323544635467042022952173625, 6.43679449975526542026435018487, 7.49024027339887201227457052610, 7.84439006039971069748622971041, 8.736150936012440857648978735320
