| L(s) = 1 | + 3.82·3-s − 3.31·5-s − 35.2·7-s − 12.3·9-s + 14.7·11-s + 13·13-s − 12.6·15-s + 111.·17-s + 6.84·19-s − 134.·21-s − 37.2·23-s − 114.·25-s − 150.·27-s − 81.7·29-s + 87.4·31-s + 56.2·33-s + 116.·35-s + 159.·37-s + 49.7·39-s + 266.·41-s + 377.·43-s + 40.9·45-s + 344.·47-s + 896.·49-s + 425.·51-s − 241.·53-s − 48.6·55-s + ⋯ |
| L(s) = 1 | + 0.735·3-s − 0.296·5-s − 1.90·7-s − 0.458·9-s + 0.402·11-s + 0.277·13-s − 0.217·15-s + 1.58·17-s + 0.0826·19-s − 1.39·21-s − 0.337·23-s − 0.912·25-s − 1.07·27-s − 0.523·29-s + 0.506·31-s + 0.296·33-s + 0.562·35-s + 0.706·37-s + 0.204·39-s + 1.01·41-s + 1.34·43-s + 0.135·45-s + 1.06·47-s + 2.61·49-s + 1.16·51-s − 0.624·53-s − 0.119·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.713938682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713938682\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 - 3.82T + 27T^{2} \) |
| 5 | \( 1 + 3.31T + 125T^{2} \) |
| 7 | \( 1 + 35.2T + 343T^{2} \) |
| 11 | \( 1 - 14.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.84T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 87.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 159.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 344.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 241.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 462.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 76.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 977.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 399.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 563.T + 9.12e5T^{2} \) |
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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
−9.494071423249406515149704959140, −9.288959718352240814527474727985, −8.072967719750406782127920449824, −7.41839005505636001334510489505, −6.23254415807738024890309024560, −5.71665373413486153644468119057, −3.93287498576658859422093018544, −3.38778148136175459758569924526, −2.48934439500396542828823265443, −0.67944344128939206431948126866,
0.67944344128939206431948126866, 2.48934439500396542828823265443, 3.38778148136175459758569924526, 3.93287498576658859422093018544, 5.71665373413486153644468119057, 6.23254415807738024890309024560, 7.41839005505636001334510489505, 8.072967719750406782127920449824, 9.288959718352240814527474727985, 9.494071423249406515149704959140
