| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s + 6·11-s + 12-s − 5·13-s + 14-s + 16-s + 3·17-s + 2·18-s − 19-s − 21-s − 6·22-s − 3·23-s − 24-s − 5·25-s + 5·26-s − 5·27-s − 28-s − 9·29-s + 4·31-s − 32-s + 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 1.80·11-s + 0.288·12-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.218·21-s − 1.27·22-s − 0.625·23-s − 0.204·24-s − 25-s + 0.980·26-s − 0.962·27-s − 0.188·28-s − 1.67·29-s + 0.718·31-s − 0.176·32-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7016851507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7016851507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.09856472798290, −13.56639139980168, −12.89916159285587, −12.32309967625486, −11.76829971400069, −11.61987638935613, −11.05354874780792, −10.11651173549817, −9.780783475964418, −9.505193476946141, −8.986137425295702, −8.426953774876378, −7.951907806455181, −7.357388851214493, −7.005544672925941, −6.097344307994004, −6.033607854491681, −5.172089943283222, −4.374359330727821, −3.735503624223504, −3.289309465810118, −2.602253464024178, −1.906685500113735, −1.427002295764017, −0.2814509226632620,
0.2814509226632620, 1.427002295764017, 1.906685500113735, 2.602253464024178, 3.289309465810118, 3.735503624223504, 4.374359330727821, 5.172089943283222, 6.033607854491681, 6.097344307994004, 7.005544672925941, 7.357388851214493, 7.951907806455181, 8.426953774876378, 8.986137425295702, 9.505193476946141, 9.780783475964418, 10.11651173549817, 11.05354874780792, 11.61987638935613, 11.76829971400069, 12.32309967625486, 12.89916159285587, 13.56639139980168, 14.09856472798290
