L(s) = 1 | − 0.414·3-s + 5-s − 2.82·7-s − 2.82·9-s + 2.41·11-s − 1.82·13-s − 0.414·15-s − 4.82·17-s + 6·19-s + 1.17·21-s + 7.65·23-s − 4·25-s + 2.41·27-s − 29-s + 4.07·31-s − 0.999·33-s − 2.82·35-s + 4·37-s + 0.757·39-s + 12.4·41-s + 6.41·43-s − 2.82·45-s − 5.24·47-s + 1.00·49-s + 1.99·51-s + 7.48·53-s + 2.41·55-s + ⋯ |
L(s) = 1 | − 0.239·3-s + 0.447·5-s − 1.06·7-s − 0.942·9-s + 0.727·11-s − 0.507·13-s − 0.106·15-s − 1.17·17-s + 1.37·19-s + 0.255·21-s + 1.59·23-s − 0.800·25-s + 0.464·27-s − 0.185·29-s + 0.731·31-s − 0.174·33-s − 0.478·35-s + 0.657·37-s + 0.121·39-s + 1.94·41-s + 0.978·43-s − 0.421·45-s − 0.764·47-s + 0.142·49-s + 0.280·51-s + 1.02·53-s + 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.341719014\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341719014\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 7.65T + 23T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 - 7.48T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + 0.828T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 0.414T + 79T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.401658744292287139220834037954, −8.686917852713124367066177778957, −7.52335098403434400317678761332, −6.74306468339347360694116807785, −6.06979600340209920629982133888, −5.36219824841842964527169089397, −4.30530291396614631505460952768, −3.17118593780715211241432796765, −2.44336144083430554859731658728, −0.78116509433349333725815648939,
0.78116509433349333725815648939, 2.44336144083430554859731658728, 3.17118593780715211241432796765, 4.30530291396614631505460952768, 5.36219824841842964527169089397, 6.06979600340209920629982133888, 6.74306468339347360694116807785, 7.52335098403434400317678761332, 8.686917852713124367066177778957, 9.401658744292287139220834037954