| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 0.723·7-s − 8-s + 9-s + 5.51·11-s + 12-s + 4.96·13-s − 0.723·14-s + 16-s + 4.23·17-s − 18-s + 4.96·19-s + 0.723·21-s − 5.51·22-s − 23-s − 24-s − 4.96·26-s + 27-s + 0.723·28-s − 29-s + 6·31-s − 32-s + 5.51·33-s − 4.23·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.273·7-s − 0.353·8-s + 0.333·9-s + 1.66·11-s + 0.288·12-s + 1.37·13-s − 0.193·14-s + 0.250·16-s + 1.02·17-s − 0.235·18-s + 1.13·19-s + 0.157·21-s − 1.17·22-s − 0.208·23-s − 0.204·24-s − 0.973·26-s + 0.192·27-s + 0.136·28-s − 0.185·29-s + 1.07·31-s − 0.176·32-s + 0.959·33-s − 0.726·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.314603756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.314603756\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 0.723T + 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 0.961T + 43T^{2} \) |
| 47 | \( 1 + 7.96T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 9.02T + 59T^{2} \) |
| 61 | \( 1 - 7.44T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 0.514T + 71T^{2} \) |
| 73 | \( 1 + 0.447T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 2.55T + 97T^{2} \) |
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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
−8.536940331286590779488832456996, −8.074318729330489632561754665292, −7.25149410515230984303213448128, −6.46530410064055042932591003558, −5.84799120812053353190741778061, −4.66168666350592162631283684863, −3.61571944705244525521802152209, −3.15038335409568857738317489071, −1.62275548728107336960943558507, −1.14556799681141627519721567645,
1.14556799681141627519721567645, 1.62275548728107336960943558507, 3.15038335409568857738317489071, 3.61571944705244525521802152209, 4.66168666350592162631283684863, 5.84799120812053353190741778061, 6.46530410064055042932591003558, 7.25149410515230984303213448128, 8.074318729330489632561754665292, 8.536940331286590779488832456996
