| L(s) = 1 | − 3·3-s + 5.71·5-s + 19.4·7-s + 9·9-s − 41.2·11-s − 17.1·15-s − 57.2·17-s + 143.·19-s − 58.3·21-s − 111.·23-s − 92.2·25-s − 27·27-s + 204.·29-s − 186.·31-s + 123.·33-s + 111.·35-s − 123.·37-s + 481.·41-s + 346.·43-s + 51.4·45-s + 446.·47-s + 35.5·49-s + 171.·51-s + 27.8·53-s − 235.·55-s − 429.·57-s − 25.0·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.511·5-s + 1.05·7-s + 0.333·9-s − 1.12·11-s − 0.295·15-s − 0.817·17-s + 1.72·19-s − 0.606·21-s − 1.00·23-s − 0.738·25-s − 0.192·27-s + 1.30·29-s − 1.08·31-s + 0.652·33-s + 0.537·35-s − 0.549·37-s + 1.83·41-s + 1.22·43-s + 0.170·45-s + 1.38·47-s + 0.103·49-s + 0.471·51-s + 0.0722·53-s − 0.577·55-s − 0.997·57-s − 0.0553·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.007683250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007683250\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 5.71T + 125T^{2} \) |
| 7 | \( 1 - 19.4T + 343T^{2} \) |
| 11 | \( 1 + 41.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 57.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 123.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 346.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 446.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 27.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 25.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 281.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 697.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 313.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 898.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 478.T + 9.12e5T^{2} \) |
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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
−8.843134951103358813085755394062, −7.68367701680158180878909139073, −7.54031829715629932959067318466, −6.20235358936932182083866158179, −5.55079913229385548624674053924, −4.93084281645911389088866972826, −4.05654766937754588713525160698, −2.67234407017855341072632431961, −1.80919262834160468132105453855, −0.67527492141298269966515562139,
0.67527492141298269966515562139, 1.80919262834160468132105453855, 2.67234407017855341072632431961, 4.05654766937754588713525160698, 4.93084281645911389088866972826, 5.55079913229385548624674053924, 6.20235358936932182083866158179, 7.54031829715629932959067318466, 7.68367701680158180878909139073, 8.843134951103358813085755394062
