| L(s) = 1 | − 5.48·3-s − 1.44·5-s − 5.95·7-s + 3.06·9-s − 34.3·11-s − 76.5·13-s + 7.90·15-s − 13.1·17-s − 45.0·19-s + 32.6·21-s + 23·23-s − 122.·25-s + 131.·27-s − 92.3·29-s − 231.·31-s + 188.·33-s + 8.59·35-s − 438.·37-s + 419.·39-s + 91.0·41-s − 415.·43-s − 4.41·45-s + 476.·47-s − 307.·49-s + 71.9·51-s + 262.·53-s + 49.4·55-s + ⋯ |
| L(s) = 1 | − 1.05·3-s − 0.129·5-s − 0.321·7-s + 0.113·9-s − 0.940·11-s − 1.63·13-s + 0.136·15-s − 0.187·17-s − 0.543·19-s + 0.339·21-s + 0.208·23-s − 0.983·25-s + 0.935·27-s − 0.591·29-s − 1.33·31-s + 0.992·33-s + 0.0414·35-s − 1.94·37-s + 1.72·39-s + 0.346·41-s − 1.47·43-s − 0.0146·45-s + 1.48·47-s − 0.896·49-s + 0.197·51-s + 0.679·53-s + 0.121·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0006944176300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0006944176300\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 5.48T + 27T^{2} \) |
| 5 | \( 1 + 1.44T + 125T^{2} \) |
| 7 | \( 1 + 5.95T + 343T^{2} \) |
| 11 | \( 1 + 34.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 92.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 231.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 91.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 685.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 543.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 840.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 195.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 337.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 605.T + 9.12e5T^{2} \) |
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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
−9.260404853004556692954883114613, −8.268852778427304724447964183647, −7.32578171171857391853342630122, −6.76385450109715637109244679418, −5.56361394173020594796069319176, −5.27808374325976344377410163808, −4.22848031566057527304563180238, −2.97914530792425302563464686971, −1.92833059935984270441551006022, −0.01274061891385634538645032817,
0.01274061891385634538645032817, 1.92833059935984270441551006022, 2.97914530792425302563464686971, 4.22848031566057527304563180238, 5.27808374325976344377410163808, 5.56361394173020594796069319176, 6.76385450109715637109244679418, 7.32578171171857391853342630122, 8.268852778427304724447964183647, 9.260404853004556692954883114613
