| L(s) = 1 | − 9·3-s − 25·5-s − 164·7-s + 81·9-s − 720·11-s + 698·13-s + 225·15-s − 2.22e3·17-s − 356·19-s + 1.47e3·21-s + 1.80e3·23-s + 625·25-s − 729·27-s + 714·29-s − 848·31-s + 6.48e3·33-s + 4.10e3·35-s − 1.13e4·37-s − 6.28e3·39-s + 9.35e3·41-s + 5.95e3·43-s − 2.02e3·45-s + 1.11e4·47-s + 1.00e4·49-s + 2.00e4·51-s + 1.41e4·53-s + 1.80e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.26·7-s + 1/3·9-s − 1.79·11-s + 1.14·13-s + 0.258·15-s − 1.86·17-s − 0.226·19-s + 0.730·21-s + 0.709·23-s + 1/5·25-s − 0.192·27-s + 0.157·29-s − 0.158·31-s + 1.03·33-s + 0.565·35-s − 1.35·37-s − 0.661·39-s + 0.869·41-s + 0.491·43-s − 0.149·45-s + 0.736·47-s + 0.600·49-s + 1.07·51-s + 0.689·53-s + 0.802·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6118980092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6118980092\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 7 | \( 1 + 164 T + p^{5} T^{2} \) |
| 11 | \( 1 + 720 T + p^{5} T^{2} \) |
| 13 | \( 1 - 698 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2226 T + p^{5} T^{2} \) |
| 19 | \( 1 + 356 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1800 T + p^{5} T^{2} \) |
| 29 | \( 1 - 714 T + p^{5} T^{2} \) |
| 31 | \( 1 + 848 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11302 T + p^{5} T^{2} \) |
| 41 | \( 1 - 9354 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5956 T + p^{5} T^{2} \) |
| 47 | \( 1 - 11160 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14106 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7920 T + p^{5} T^{2} \) |
| 61 | \( 1 + 13450 T + p^{5} T^{2} \) |
| 67 | \( 1 - 65476 T + p^{5} T^{2} \) |
| 71 | \( 1 + 34560 T + p^{5} T^{2} \) |
| 73 | \( 1 - 86258 T + p^{5} T^{2} \) |
| 79 | \( 1 - 108832 T + p^{5} T^{2} \) |
| 83 | \( 1 + 10668 T + p^{5} T^{2} \) |
| 89 | \( 1 - 10818 T + p^{5} T^{2} \) |
| 97 | \( 1 - 4418 T + p^{5} T^{2} \) |
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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
−10.89581353712377024745827803222, −10.72022391522388610886261329095, −9.344872854140805077518835704760, −8.348341694552421965485012134524, −7.08399523225577673860474131480, −6.25263771345026341743253032696, −5.10529778034645029102703644458, −3.80552022498707371377988913896, −2.51953314733531064468403921413, −0.44981945464797214779218231291,
0.44981945464797214779218231291, 2.51953314733531064468403921413, 3.80552022498707371377988913896, 5.10529778034645029102703644458, 6.25263771345026341743253032696, 7.08399523225577673860474131480, 8.348341694552421965485012134524, 9.344872854140805077518835704760, 10.72022391522388610886261329095, 10.89581353712377024745827803222
