L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 7-s − 8-s + 9-s + 2·10-s + 6·11-s − 12-s − 6·13-s + 14-s + 2·15-s + 16-s − 2·17-s − 18-s − 2·20-s + 21-s − 6·22-s − 23-s + 24-s − 25-s + 6·26-s − 27-s − 28-s − 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 1.80·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.218·21-s − 1.27·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02206995353\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02206995353\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.27973331578003, −12.07546947849299, −11.50735451436492, −11.38753957093456, −10.87070199383996, −10.02327993507832, −9.877021585074950, −9.476876240670518, −8.963184392823448, −8.427874744485629, −7.983816243493778, −7.334191974885015, −7.061565996707366, −6.711946320992748, −6.142100735505099, −5.694146262583650, −4.863767154035141, −4.593571545906587, −3.913641316960724, −3.517810516139524, −2.955763934071615, −2.026102544978200, −1.737948628955166, −0.9187158595426046, −0.05352173160283814,
0.05352173160283814, 0.9187158595426046, 1.737948628955166, 2.026102544978200, 2.955763934071615, 3.517810516139524, 3.913641316960724, 4.593571545906587, 4.863767154035141, 5.694146262583650, 6.142100735505099, 6.711946320992748, 7.061565996707366, 7.334191974885015, 7.983816243493778, 8.427874744485629, 8.963184392823448, 9.476876240670518, 9.877021585074950, 10.02327993507832, 10.87070199383996, 11.38753957093456, 11.50735451436492, 12.07546947849299, 12.27973331578003