| L(s) = 1 | − 3·3-s − 21.3·5-s + 9·9-s − 10.7·11-s + 8.31·13-s + 64.1·15-s + 1.48·17-s − 161.·19-s + 123.·23-s + 332.·25-s − 27·27-s − 222.·29-s − 203.·31-s + 32.1·33-s − 341.·37-s − 24.9·39-s + 147.·41-s − 51.3·43-s − 192.·45-s − 269.·47-s − 4.45·51-s − 432.·53-s + 229.·55-s + 484.·57-s − 30.7·59-s − 592.·61-s − 177.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.91·5-s + 0.333·9-s − 0.293·11-s + 0.177·13-s + 1.10·15-s + 0.0211·17-s − 1.95·19-s + 1.12·23-s + 2.66·25-s − 0.192·27-s − 1.42·29-s − 1.18·31-s + 0.169·33-s − 1.51·37-s − 0.102·39-s + 0.560·41-s − 0.182·43-s − 0.637·45-s − 0.836·47-s − 0.0122·51-s − 1.12·53-s + 0.561·55-s + 1.12·57-s − 0.0678·59-s − 1.24·61-s − 0.339·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.09472746363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09472746363\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 21.3T + 125T^{2} \) |
| 11 | \( 1 + 10.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.31T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.48T + 4.91e3T^{2} \) |
| 19 | \( 1 + 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 147.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 51.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 432.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 30.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 592.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 414.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 594.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 831.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 326.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.571309853931210706013076769132, −7.76721106078762164690071259782, −7.17637435402170587952074238053, −6.46960078999102448038673613975, −5.35685417050769085681830136305, −4.53252236737547542986776523613, −3.89276502332063731568236633323, −3.07396242464859802904135343167, −1.60249598878617700424103149091, −0.14100279678550254621417585633,
0.14100279678550254621417585633, 1.60249598878617700424103149091, 3.07396242464859802904135343167, 3.89276502332063731568236633323, 4.53252236737547542986776523613, 5.35685417050769085681830136305, 6.46960078999102448038673613975, 7.17637435402170587952074238053, 7.76721106078762164690071259782, 8.571309853931210706013076769132
