L(s) = 1 | + 0.414·2-s − 2.82·3-s − 1.82·4-s − 1.17·6-s + 0.414·7-s − 1.58·8-s + 5.00·9-s + 3.58·11-s + 5.17·12-s − 13-s + 0.171·14-s + 3·16-s − 1.82·17-s + 2.07·18-s + 7.65·19-s − 1.17·21-s + 1.48·22-s + 8.82·23-s + 4.48·24-s − 0.414·26-s − 5.65·27-s − 0.757·28-s − 5.82·29-s − 7.24·31-s + 4.41·32-s − 10.1·33-s − 0.757·34-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 1.63·3-s − 0.914·4-s − 0.478·6-s + 0.156·7-s − 0.560·8-s + 1.66·9-s + 1.08·11-s + 1.49·12-s − 0.277·13-s + 0.0458·14-s + 0.750·16-s − 0.443·17-s + 0.488·18-s + 1.75·19-s − 0.255·21-s + 0.316·22-s + 1.84·23-s + 0.915·24-s − 0.0812·26-s − 1.08·27-s − 0.143·28-s − 1.08·29-s − 1.30·31-s + 0.780·32-s − 1.76·33-s − 0.129·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7425311965\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7425311965\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 0.414T + 7T^{2} \) |
| 11 | \( 1 - 3.58T + 11T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 4.75T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 + 4.82T + 97T^{2} \) |
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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
−11.45255025936690215557238566624, −11.11477908981489042514208322709, −9.661693432332869129548718238911, −9.169947437687977848265208195093, −7.53607462685484137032843166972, −6.51374032857296880535367423457, −5.44217524763044622936212804396, −4.86658875375843851726464671663, −3.66338328472537411261628031991, −0.951268133424611537112902673213,
0.951268133424611537112902673213, 3.66338328472537411261628031991, 4.86658875375843851726464671663, 5.44217524763044622936212804396, 6.51374032857296880535367423457, 7.53607462685484137032843166972, 9.169947437687977848265208195093, 9.661693432332869129548718238911, 11.11477908981489042514208322709, 11.45255025936690215557238566624