L(s) = 1 | + 238.·2-s − 5.90e4·3-s − 2.04e6·4-s − 1.41e7·6-s + 6.22e8·7-s − 9.88e8·8-s + 3.48e9·9-s + 1.17e10·11-s + 1.20e11·12-s + 6.64e11·13-s + 1.48e11·14-s + 4.04e12·16-s + 1.24e13·17-s + 8.33e11·18-s + 1.38e13·19-s − 3.67e13·21-s + 2.81e12·22-s + 2.44e14·23-s + 5.83e13·24-s + 1.58e14·26-s − 2.05e14·27-s − 1.27e15·28-s − 2.45e15·29-s + 6.56e15·31-s + 3.03e15·32-s − 6.94e14·33-s + 2.96e15·34-s + ⋯ |
L(s) = 1 | + 0.164·2-s − 0.577·3-s − 0.972·4-s − 0.0952·6-s + 0.833·7-s − 0.325·8-s + 0.333·9-s + 0.136·11-s + 0.561·12-s + 1.33·13-s + 0.137·14-s + 0.919·16-s + 1.49·17-s + 0.0549·18-s + 0.519·19-s − 0.481·21-s + 0.0225·22-s + 1.23·23-s + 0.187·24-s + 0.220·26-s − 0.192·27-s − 0.810·28-s − 1.08·29-s + 1.43·31-s + 0.477·32-s − 0.0789·33-s + 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.421177105\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.421177105\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 238.T + 2.09e6T^{2} \) |
| 7 | \( 1 - 6.22e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.17e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 6.64e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.24e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.38e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.44e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.45e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.56e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 8.38e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.59e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.03e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 6.98e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.25e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.40e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.88e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.90e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 6.30e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.29e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 5.72e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.21e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.43e20T + 5.27e41T^{2} \) |
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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
−10.72392482444686728280287720967, −9.566963582020029218897975577725, −8.521361551532477269354594592420, −7.54286025231621335173474401711, −5.98026570368909040230774574597, −5.23003213626165196844630892723, −4.23140523753069354951477624553, −3.20736019544651136488502175940, −1.31479010177546473980100264157, −0.78850668214242117767840919149,
0.78850668214242117767840919149, 1.31479010177546473980100264157, 3.20736019544651136488502175940, 4.23140523753069354951477624553, 5.23003213626165196844630892723, 5.98026570368909040230774574597, 7.54286025231621335173474401711, 8.521361551532477269354594592420, 9.566963582020029218897975577725, 10.72392482444686728280287720967