| L(s) = 1 | + 3·3-s − 25·5-s − 49·7-s − 234·9-s − 405·11-s − 391·13-s − 75·15-s + 999·17-s − 2.34e3·19-s − 147·21-s − 2.43e3·23-s + 625·25-s − 1.43e3·27-s + 8.25e3·29-s − 4.01e3·31-s − 1.21e3·33-s + 1.22e3·35-s − 7.04e3·37-s − 1.17e3·39-s + 3.33e3·41-s + 2.35e4·43-s + 5.85e3·45-s − 1.03e4·47-s + 2.40e3·49-s + 2.99e3·51-s + 3.08e3·53-s + 1.01e4·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.447·5-s − 0.377·7-s − 0.962·9-s − 1.00·11-s − 0.641·13-s − 0.0860·15-s + 0.838·17-s − 1.48·19-s − 0.0727·21-s − 0.957·23-s + 1/5·25-s − 0.377·27-s + 1.82·29-s − 0.750·31-s − 0.194·33-s + 0.169·35-s − 0.845·37-s − 0.123·39-s + 0.309·41-s + 1.93·43-s + 0.430·45-s − 0.681·47-s + 1/7·49-s + 0.161·51-s + 0.150·53-s + 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8672905001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8672905001\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - p T + p^{5} T^{2} \) |
| 11 | \( 1 + 405 T + p^{5} T^{2} \) |
| 13 | \( 1 + 391 T + p^{5} T^{2} \) |
| 17 | \( 1 - 999 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2342 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2430 T + p^{5} T^{2} \) |
| 29 | \( 1 - 8259 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4016 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7042 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3336 T + p^{5} T^{2} \) |
| 43 | \( 1 - 23518 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10317 T + p^{5} T^{2} \) |
| 53 | \( 1 - 3084 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18816 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21668 T + p^{5} T^{2} \) |
| 67 | \( 1 + 52124 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28560 T + p^{5} T^{2} \) |
| 73 | \( 1 + 70342 T + p^{5} T^{2} \) |
| 79 | \( 1 + 58823 T + p^{5} T^{2} \) |
| 83 | \( 1 + 756 T + p^{5} T^{2} \) |
| 89 | \( 1 - 135384 T + p^{5} T^{2} \) |
| 97 | \( 1 - 110435 T + p^{5} T^{2} \) |
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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
−10.15192115795869857838353514070, −8.948314167369957459276500043997, −8.208200243308395796302434846113, −7.47129907455081247803957419187, −6.28922551278524797523438370732, −5.39751006135097991580984639624, −4.28094725904091955264225850362, −3.09567909604928904139392292701, −2.26475194427441717514435378588, −0.42496416859401474821204349671,
0.42496416859401474821204349671, 2.26475194427441717514435378588, 3.09567909604928904139392292701, 4.28094725904091955264225850362, 5.39751006135097991580984639624, 6.28922551278524797523438370732, 7.47129907455081247803957419187, 8.208200243308395796302434846113, 8.948314167369957459276500043997, 10.15192115795869857838353514070
