| L(s) = 1 | − 12·3-s − 88·7-s − 99·9-s − 540·11-s + 418·13-s − 594·17-s − 836·19-s + 1.05e3·21-s − 4.10e3·23-s + 4.10e3·27-s − 594·29-s − 4.25e3·31-s + 6.48e3·33-s + 298·37-s − 5.01e3·39-s + 1.72e4·41-s − 1.21e4·43-s − 1.29e3·47-s − 9.06e3·49-s + 7.12e3·51-s − 1.94e4·53-s + 1.00e4·57-s + 7.66e3·59-s − 3.47e4·61-s + 8.71e3·63-s + 2.18e4·67-s + 4.92e4·69-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.678·7-s − 0.407·9-s − 1.34·11-s + 0.685·13-s − 0.498·17-s − 0.531·19-s + 0.522·21-s − 1.61·23-s + 1.08·27-s − 0.131·29-s − 0.795·31-s + 1.03·33-s + 0.0357·37-s − 0.528·39-s + 1.60·41-s − 0.997·43-s − 0.0855·47-s − 0.539·49-s + 0.383·51-s − 0.953·53-s + 0.408·57-s + 0.286·59-s − 1.19·61-s + 0.276·63-s + 0.593·67-s + 1.24·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4951380905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4951380905\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 88 T + p^{5} T^{2} \) |
| 11 | \( 1 + 540 T + p^{5} T^{2} \) |
| 13 | \( 1 - 418 T + p^{5} T^{2} \) |
| 17 | \( 1 + 594 T + p^{5} T^{2} \) |
| 19 | \( 1 + 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 594 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 - 298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 + 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 - 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 - 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 - 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 + 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 - 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 - 122398 T + p^{5} T^{2} \) |
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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
−10.64904729172115075975351157055, −9.703144930712853659992587894501, −8.574672049114380686164620451120, −7.69272570469616365474407029740, −6.36966168520831904151986313229, −5.84012022924798646875943307799, −4.74613222887416704278550168572, −3.42039788671378080144164335057, −2.17475669847966030665475965789, −0.36361852741090987730834854544,
0.36361852741090987730834854544, 2.17475669847966030665475965789, 3.42039788671378080144164335057, 4.74613222887416704278550168572, 5.84012022924798646875943307799, 6.36966168520831904151986313229, 7.69272570469616365474407029740, 8.574672049114380686164620451120, 9.703144930712853659992587894501, 10.64904729172115075975351157055
